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determinant of a 2 " }{TEXT 273 1 "x" }{TEXT -1 10 " 2 matrix " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The determinant of a 2 " }{TEXT 274 1 "x" }{TEXT -1 10 " 2 matr ix " }{XPPEDIT 18 0 "A=matrix([[a, b], [c, d]])" "6#/%\"AG-%'matrixG6# 7$7$%\"aG%\"bG7$%\"cG%\"dG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 238 93 93 {PLOTDATA 2 "6/-%'CURVESG6$7$7$$\" 33+++++++E!#<$!35+++++++?!#=7$F($\"3%**************>\"F*-%'COLOURG6&%$ RGBG\"\"!F5F5-F$6$7$7$$\"3M+++++++WF*F+7$F:F/F1-%%TEXTG6&7$$\"\"$F5$\" \"\"F5Q\"a6\"F1-%%FONTG6%%&TIMESG%'ITALICG\"#7-F>6&7$$\"\"%F5FCQ\"bFFF 1FG-F>6&7$FA$F5F5Q\"cFFF1FG-F>6&7$FPFVQ\"dFFF1FG-F>6&7$$\"$#=!\"#$\"#` F[oQ\"AFFF1FG-F>6&7$$\"$x%F[o$\"\"&!\"\"Q\"=FFF1FG-F>6&7$$\"#cFfoF\\oQ *a~d~-~b~cFFF1FG-F>6&7$$\"$x\"F[oF\\oQ+det(~~~)~=FFF1-FH6%FJ%&ROMANGFL -%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FFF^q-%%VIEWG6$%(DEFAULTGFbq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "The \+ inverse matrix " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" } {TEXT -1 4 " of " }{TEXT 275 1 "A" }{TEXT -1 13 " exists when " } {XPPEDIT 18 0 "det(A)<>0" "6#0-%$detG6#%\"AG\"\"!" }{TEXT -1 5 " and \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A^(-1)=1/det(A)" "6#/)%\"AG,$\"\"\"!\"\"*&F'F'-%$detG6#F%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[d,-b], [-c,a]])" "6#-%'matrixG6#7$7$%\"dG,$%\"bG!\"\"7$ ,$%\"cGF+%\"aG" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The determinant of \+ a 3 " }{TEXT 272 1 "x" }{TEXT -1 10 " 3 matrix " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The determin ant of a 3 " }{TEXT 263 1 "x" }{TEXT -1 10 " 3 matrix " }{XPPEDIT 18 0 "A = matrix([[a, b, c], [d, e, f], [g, h, k]]);" "6#/%\"AG-%'matrixG 6#7%7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG7%%\"gG%\"hG%\"kG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 489 111 111 {PLOTDATA 2 "6H-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+ ++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F +7$F:F.F1-F$6$7$7$$\"3k*************4%F0$\"3))**************HF*7$FA$\" 33+++++++;F0F1-F$6$7$7$$\"3M+++++++fF0FC7$FLFFF1-F$6$7$7$$\"3k******** *****4(F0FC7$FSFFF1-F$6$7$7$$\"3M+++++++*)F0FC7$FZFFF1-F$6$7$7$$\"3(** ***********45!#;FC7$F[oFFF1-F$6$7$7$$\"3/++++++!>\"F]oFC7$FcoFFF1-%%TE XTG6&7$$F5F5$\"\"#F5Q\"a6\"F1-%%FONTG6%%&TIMESG%'ITALICG\"#7-Fgo6&7$$ \"\"\"F5F[pQ\"bF^pF1F_p-Fgo6&7$F[pF[pQ\"cF^pF1F_p-Fgo6&7$FjoFhpQ\"dF^p F1F_p-Fgo6&7$FhpFhpQ\"eF^pF1F_p-Fgo6&7$F[pFhpQ\"fF^pF1F_p-Fgo6&7$FjoFj oQ\"gF^pF1F_p-Fgo6&7$FhpFjoQ\"hF^pF1F_p-Fgo6&7$F[pFjoQ\"kF^pF1F_p-Fgo6 &7$$\"#X!\"\"$\"#:F\\sFfqF1F_p-Fgo6&7$$\"#bF\\sF]sFjqF1F_p-Fgo6&7$Fjr$ \"\"&F\\sFbrF1F_p-Fgo6&7$FbsFgsFfrF1F_p-Fgo6&7$$\"#vF\\sF]sFbqF1F_p-Fg o6&7$$\"#&)F\\sF]sFjqF1F_p-Fgo6&7$F_tFgsF^rF1F_p-Fgo6&7$FdtFgsFfrF1F_p -Fgo6&7$$\"$0\"F\\sF]sFbqF1F_p-Fgo6&7$$\"$:\"F\\sF]sFfqF1F_p-Fgo6&7$F_ uFgsF^rF1F_p-Fgo6&7$FduFgsFbrF1F_p-Fgo6&7$$!$U\"!\"#FhpQ\"AF^pF1F_p-Fg o6&7$$\"#JF\\s$\"#'*FavQ\"=F^pF1F_p-Fgo6&7$$\"#PF\\sFhpF]pF1F_p-Fgo6&7 $$\"#mF\\sFhpQ$-~bF^pF1F_p-Fgo6&7$$FivF\\sFhpQ$+~cF^pF1F_p-Fgo6&7$$!#: F\\sFhpQ+det(~~~)~=F^pF1-F`p6%Fbp%&ROMANGFdp-%*AXESSTYLEG6#%%NONEG-%+A XESLABELSG6$Q!F^pF[y-%%VIEWG6$%(DEFAULTGF_y" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=a*(e*k- f*h)-b*(d*k-f*g)+c*(d*h-e*g)" "6#/%!G,(*&%\"aG\"\"\",&*&%\"eGF(%\"kGF( F(*&%\"fGF(%\"hGF(!\"\"F(F(*&%\"bGF(,&*&%\"dGF(F,F(F(*&F.F(%\"gGF(F0F( F0*&%\"cGF(,&*&F5F(F/F(F(*&F+F(F7F(F0F(F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=a*e*k-a*f*h-b*d*k+b*f*g+c*d*h-c*e*g" "6#/%!G,.*(%\"aG\"\"\"%\"eG F(%\"kGF(F(*(F'F(%\"fGF(%\"hGF(!\"\"*(%\"bGF(%\"dGF(F*F(F.*(F0F(F,F(% \"gGF(F(*(%\"cGF(F1F(F-F(F(*(F5F(F)F(F3F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "This method of \+ evaluation is called evaluation by " }{TEXT 259 45 "expanding the dete rminant along the first row" }{TEXT -1 27 " consisting of the entries \+ " }{TEXT 266 1 "a" }{TEXT -1 2 ", " }{TEXT 267 1 "b" }{TEXT -1 5 " and " }{TEXT 268 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 125 "The value of det(A) is the sum of three terms each of which is plus or minus an entry from the first row mult iplied by the 2 " }{TEXT 264 1 "x" }{TEXT -1 57 " 2 determinant formed from the entries of the original 3 " }{TEXT 265 1 "x" }{TEXT -1 84 " \+ 3 determinant which remain when the row and column containing the part icular entry " }{TEXT 269 1 "a" }{TEXT -1 2 ", " }{TEXT 270 1 "b" } {TEXT -1 4 " or " }{TEXT 271 1 "c" }{TEXT -1 37 " from the first row a re both deleted." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 314 134 134 {PLOTDATA 2 "6:-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$ F($\"33+++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!*********** ***R#F0F+7$F:F.F1-F$6$7$7$$\"3k*************f%F0$\"3))**************HF *7$FA$\"33+++++++;F0F1-F$6$7$7$$\"3M+++++++kF0FC7$FLFFF1-F$6&7$7$$\"#F !\"\"$\"\"\"F57$$\"#PFUFV7%7$$\"++++IN!\"*$\"++++l5FinFX7$Fgn$\"++++]$ *!#5-%&STYLEG6#%,PATCHNOGRIDG-F26&F4$\"*++++\"!\")$F5F5Fio-%)POLYGONSG 6&7&7$$!\"#FUF_p7$F_p$\"#@FU7$$\"\"#FUFbp7$FepF_p7&Fap7$$\"#AFUFbp7$Fj p$\"#=FU7$F_pF]q-%&COLORG6&F4$\"\"*FUFcqFcqF`o-F$6$777$$\"35+++++++?F* $\"3'*************\\>F07$$\"3&******H.8@!>F*$\"3;+++*R.=,#F07$$\"31+++ *)R.=;F*$\"3++++]qbn?F07$$\"3-+++/0dv6F*$\"3!)******)R.=6#F07$$\"3I+++ w)R.='!#>$\"32+++.8@S@F07$$!3;+++;w1-T!#G$\"3#*************\\@F07$$!3E +++%))R.='F_sF`s7$$!3'******\\]qb<\"F*Fjr7$$!31+++*)R.=;F*Fer7$$!3&*** ***H.8@!>F*F`r7$F+F[r7$Fbt$\"3)******4g'>))=F07$$!3)******z)R.=;F*$\"3 /+++\\HWK=F07$$!3-+++/0dv6F*$\"35+++,m>)y\"F07$$!3K+++o)R.='F_s$\"33++ +(p)yf\"F0$\"3 '*************\\>F07$$\"35+++.8@!>\"F0$\"3;+++*R.=,#F07$$\"33+++*R.=; \"F0$\"3++++]qbn?F07$$\"30+++]qb<6F0$\"3!)******)R.=6#F07$$\"3)******* )R.=1\"F0$\"32+++.8@S@F07$FV$\"3#*************\\@F07$$\"3O+++7g'>Q*F*F bs7$$\"3=+++&\\HW#))F*F]s7$$\"3]+++6g'>Q)F*Fhr7$$\"3w*****p'p)y4)F*Fcr 7$$\"3U+++++++!)F*F^r7$Fat$\"3)******4g'>))=F07$$\"3Y*****>,m>Q)F*$\"3 /+++\\HWK=F07$$\"3E+++'\\HW#))F*$\"35+++,m>)y\"F07$$\"3W+++8g'>Q*F*$\" 33+++(p)yfF07$$\"3))*****HI6->#F0$\"3;++ +*R.=,#F07$$\"33+++*R.=;#F0$\"3++++]qbn?F07$$\"3$)******\\qb<@F0$\"3!) ******)R.=6#F07$$\"3)*******)R.=1#F0$\"32+++.8@S@F07$$\"\"#F5$\"3#**** *********\\@F07$$\"3-+++,m>Q>F0F]s7$$\"3%*******\\HW#)=F0Fhr7$$\"3$*** ***4g'>Q=F0Fcr7$$\"3*)*****pp)y4=F0F^r7$$\"3/+++++++=F0Fiq7$F^t$\"3)** ****4g'>))=F07$F[t$\"3/+++\\HWK=F07$Fhs$\"35+++,m>)y\"F07$Fes$\"33+++( p)yf6&7$$\"\"\"F5FBQ\"-FEF1FF-F>6&7$FBFBFDF1FF -F>6&7$FAFOFQF1FF-F>6&7$FOFOFDF1FF-F>6&7$FBFOFQF1FF-F>6&7$FAFAFDF1FF-F >6&7$FOFAFQF1FF-F>6&7$FBFAFDF1FF-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$ Q!FEFho-%%VIEWG6$%(DEFAULTGF\\p" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "For example, the determi nant can be " }{TEXT 259 31 "expanded down the second column" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 " The method just given can be extended to 4 x 4 determinants and higher dimensional determinants. " }}{PARA 0 "" 0 "" {TEXT -1 95 "A special \+ method which works just for 3 x 3 determinants is suggested by the fol lowing scheme. " }}{PARA 0 "" 0 "" {TEXT -1 82 "The first and 2nd colu mns are rewritten in order to the right of the determinant. " }}{PARA 0 "" 0 "" {TEXT -1 132 "Products of three entries following the six di agonal arrows are formed. The products corresponding to the three down ward arrows are " }{TEXT 260 5 "added" }{TEXT -1 72 " together, and th en the products corresponding to the upward arrows are " }{TEXT 256 10 "subtracted" }{TEXT -1 39 " to give the value of the determinant. \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 334 218 218 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++@!#< -%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F+7$F:F.F1- F$6(7$7$$!\"\"!\"#FA7$$\"$q#FCFE7%7$$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7%7$$\"+\"* pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\"\"$FBF^rFYFg n-F$6(7$7$F]oF]p7$F`oF^q7%7$$\"+/gN\"F]oFC7$FcoFFF1-%%TEXTG6&7$$F5 F5$\"\"#F5Q\"36\"F1-%%FONTG6%%&TIMESG%&ROMANG\"#7-Fgo6&7$$\"\"\"F5F[pQ \"5F^pF1F_p-Fgo6&7$F[pF[pQ#-6F^pF1F_p-Fgo6&7$FjoFhpQ\"2F^pF1F_p-Fgo6&7 $FhpFhpQ\"4F^pF1F_p-Fgo6&7$F[pFhpQ\"7F^pF1F_p-Fgo6&7$FjoFjoQ\"8F^pF1F_ p-Fgo6&7$FhpFjoQ#-2F^pF1F_p-Fgo6&7$F[pFjoQ\"9F^pF1F_p-Fgo6&7$$\"#X!\" \"$\"#:F\\sFfqF1F_p-Fgo6&7$$\"#bF\\sF]sFjqF1F_p-Fgo6&7$Fjr$\"\"&F\\sFb rF1F_p-Fgo6&7$FbsFgsFfrF1F_p-Fgo6&7$$\"#vF\\sF]sFbqF1F_p-Fgo6&7$$\"#&) F\\sF]sFjqF1F_p-Fgo6&7$F_tFgsF^rF1F_p-Fgo6&7$FdtFgsFfrF1F_p-Fgo6&7$$\" $0\"F\\sF]sFbqF1F_p-Fgo6&7$$\"$:\"F\\sF]sFfqF1F_p-Fgo6&7$F_uFgsF^rF1F_ p-Fgo6&7$FduFgsFbrF1F_p-Fgo6&7$$\"#PF\\sFhpF]pF1F_p-Fgo6&7$$\"#nF\\sFh pFjpF1F_p-Fgo6&7$$\"#(*F\\sFhpQ\"6F^pF1F_p-Fgo6&7$$\"#JF\\s$\"#'*!\"#Q \"=F^pF1F_p-Fgo6&7$$\"$D'FcwFawQ\"-F^pF1F_p-Fgo6&7$$\"$D*FcwFawFjwF1F_ p-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F^pFgx-%%VIEWG6$%(DEFAULTGF[y " 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+ 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur ve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 3 4" "Curve 35" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 3*(36+14)-5*(18-56)-6*(-4-32);" "6#/%!G,(*&\"\"$\" \"\",&\"#OF(\"#9F(F(F(*&\"\"&F(,&\"#=F(\"#c!\"\"F(F1*&\"\"'F(,&\"\"%F1 \"#KF1F(F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=3*50-5*(-38)-6*(-36)" "6 #/%!G,(*&\"\"$\"\"\"\"#]F(F(*&\"\"&F(,$\"#Q!\"\"F(F.*&\"\"'F(,$\"#OF.F (F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=150+190+216" "6#/%!G,(\"$]\"\"\" \"\"$!>F'\"$;#F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=556" "6#/%!G\"$c& " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "The diagonal method \+ gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 375 229 229 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+ ++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F +7$F:F.F1-F$6(7$7$$!\"\"!\"#FA7$$\"$q#FCFE7%7$$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7 %7$$\"+\"*pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\" \"$FBF^rFYFgn-F$6(7$7$F]oF]p7$F`oF^q7%7$$\"+/gNF%\"#UF%\"#!*!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=388+216-48 " "6#/%!G,(\"$)Q\"\"\"\"$;#F'\"#[!\"\"" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=388+168" "6#/%!G,&\"$)Q\"\"\" \"$o\"F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=556" "6#/%!G\"$c&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "matrix([[3 ,5,-6],[2,4,7],[8,-2,9]]);\nlinalg[det](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"$\"\"&!\"'7%\"\"#\"\"%\"\"(7%\"\")! \"#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$c&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "The determinant of a 4 " }{TEXT 285 1 " x" }{TEXT -1 10 " 4 matrix " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "A = matrix([ [a,b,c,d],[e,f,g,h],[k,m,n,p],[q,r,s,t]])" "6#/%\"AG-%'matrixG6#7&7&% \"aG%\"bG%\"cG%\"dG7&%\"eG%\"fG%\"gG%\"hG7&%\"kG%\"mG%\"nG%\"pG7&%\"qG %\"rG%\"sG%\"tG" }{TEXT -1 61 ", then, by expansion along the first ro w, the determinant of " }{TEXT 291 1 "A" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 707 129 129 {PLOTDATA 2 "6bo-%'C URVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++J!#<-%'COLOUR 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an entry from the first row multiplied by the 3 " }{TEXT 286 1 "x" }{TEXT -1 57 " 3 determinant formed from the entries of the \+ original 4 " }{TEXT 287 1 "x" }{TEXT -1 84 " 4 determinant which remai n when the row and column containing the particular entry " }{TEXT 288 1 "a" }{TEXT -1 2 ", " }{TEXT 289 4 "b, c" }{TEXT -1 4 " or " } {TEXT 290 1 "d" }{TEXT -1 37 " from the first row are both deleted." } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 339 145 145 {PLOTDATA 2 " 6F-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++J!#<-%' COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R$F0F+7$F:F.F1-F$6 $7$7$$\"3k*************f&F0$\"3))**************HF*7$FA$\"33+++++++EF0F 1-F$6$7$7$$\"3M+++++++%)F0FC7$FLFFF1-F$6&7$7$$\"#P!\"\"$\"#:FU7$$\"#ZF UFV7%7$$\"++++IX!\"*$\"++++l:FinFX7$Fgn$\"++++N9Fin-%&STYLEG6#%,PATCHN OGRIDG-F26&F4$\"*++++\"!\")$F5F5Fho-%)POLYGONSG6&7&7$$!\"#FUF^p7$F^p$ \"#JFU7$$\"\"#FUFap7$FdpF^p7&F`p7$$\"#KFUFap7$Fip$\"#GFU7$F^pF\\q-%&CO 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>#F0$\"3;+++*R.=,$F07$$\"33+++*R.=;#F0$\"3++++]qbnIF07$$\"3$)******\\q b<@F0$\"3!)******)R.=6$F07$$\"3)*******)R.=1#F0$\"32+++.8@SJF07$$\"\"# F5$\"3#*************\\JF07$$\"3-+++,m>Q>F0Fas7$$\"3%*******\\HW#)=F0F \\s7$$\"3$******4g'>Q=F0Fgr7$$\"3*)*****pp)y4=F0Fbr7$$\"3/+++++++=F0F] r7$Fbt$\"3?+++,m>))GF07$F_t$\"3#)******[HWKGF07$F\\t$\"35+++,m>)y#F07$ Fis$\"3%)*****pp)yfFF07$Fds$\"3+++++++]FF07$F_sFau7$$\"3\"******40dv6# F0F^u7$Fer$\"3!*******\\HWKGF07$F`rFhtFjqFco-%%TEXTG6&7$Fho$\"\"$F5Q\" a6\"F1-%%FONTG6%%&TIMESG%'ITALICG\"#7-F_v6&7$$\"\"\"F5FbvQ\"bFevF1Ffv- F_v6&7$FdsFbvQ\"cFevF1Ffv-F_v6&7$FbvFbvQ\"dFevF1Ffv-F_v6&7$FhoFdsQ\"eF evF1Ffv-F_v6&7$F_wFdsQ\"fFevF1Ffv-F_v6&7$FdsFdsQ\"gFevF1Ffv-F_v6&7$Fbv FdsQ\"hFevF1Ffv-F_v6&7$FhoF_wQ\"kFevF1Ffv-F_v6&7$F_wF_wQ\"mFevF1Ffv-F_ v6&7$FdsF_wQ\"nFevF1Ffv-F_v6&7$FbvF_wQ\"pFevF1Ffv-F_v6&7$FhoFhoQ\"qFev F1Ffv-F_v6&7$F_wFhoQ\"rFevF1Ffv-F_v6&7$FdsFhoQ\"sFevF1Ffv-F_v6&7$FbvFh oQ\"tFevF1Ffv-F_v6&7$$\"\"'F5$\"#DFUF]xF1Ffv-F_v6&7$$\"\"(F5F_[lFaxF1F 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FA$\"33+++++++EF0F1-F$6$7$7$$\"3M+++++++%)F0FC7$FLFFF1-F$6&7$7$$\"#P! \"\"$\"#:FU7$$\"#ZFUFV7%7$$\"++++IX!\"*$\"++++l:FinFX7$Fgn$\"++++N9Fin -%&STYLEG6#%,PATCHNOGRIDG-F26&F4$\"*++++\"!\")$F5F5Fho-%)POLYGONSG6&7& 7$$\"#GFU$!\"#FU7$F^p$\"#JFU7$$\"#KFUFcp7$FfpF`p7&7$F`pFcpFep7$FfpF^p7 $F`pF^p-%&COLORG6&F4$\"\"*FUF`qF`qF_o-F$6$777$$\"3;+++++++KF0$\"3=++++ ++]HF07$$\"3))*****HI6->$F0$\"3;+++*R.=,$F07$$\"33+++*R.=;$F0$\"3++++] qbnIF07$$\"3$)******\\qbQHF0F\\s7 $$\"3;+++]HW#)GF0Fgr7$$\"3$******4g'>QGF0Fbr7$$\"36+++(p)y4GF0F]r7$$\" 3#)*************z#F0Fhq7$F]t$\"3?+++,m>))GF07$Fjs$\"3#)******[HWKGF07$ Fgs$\"35+++,m>)y#F07$Fds$\"3%)*****pp)yfFF07$F_s$\"3+++++++]FF07$FjrF \\u7$$\"3\"******40dv6$F0Fit7$F`r$\"3!*******\\HWKGF07$F[rFctFeqFco-%% TEXTG6&7$FhoF_sQ\"a6\"F1-%%FONTG6%%&TIMESG%'ITALICG\"#7-Fju6&7$$\"\"\" F5F_sQ\"bF^vF1F_v-Fju6&7$$\"\"#F5F_sQ\"cF^vF1F_v-Fju6&7$F_sF_sQ\"dF^vF 1F_v-Fju6&7$FhoF^wQ\"eF^vF1F_v-Fju6&7$FhvF^wQ\"fF^vF1F_v-Fju6&7$F^wF^w 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"Curve 33" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 141 "This determinant can be expanded in a similar way along any ro w or down any column provided that the following chequerboard of signs is used." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 238 172 172 {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q-%%VIEWG6$%(DEFAULTGFcq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Cur ve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 293 7 "Example" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 33 "If the determinant of the matrix " }{XPPEDIT 18 0 "A = matrix([[0, 1, 4, 5], [3, 1, 2, 1], [0, 1, 3, 0], [2, 1, 0, 3]]);" "6#/%\"AG-%'matrixG6#7&7 &\"\"!\"\"\"\"\"%\"\"&7&\"\"$F+\"\"#F+7&F*F+F/F*7&F0F+F*F/" }{TEXT -1 71 " is evaluated by expanding down the first column, there are only t wo 3 " }{TEXT 292 1 "x" }{TEXT -1 26 " 3 determinants to find. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 425 141 141 {PLOTDATA 2 "6P-%'CURVESG6$7$7$$!3A+++++++S!#=$! 35+++++++?F*7$F($\"33+++++++J!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$ \"3!**************R$F0F+7$F:F.F1-F$6$7$7$$\"3k*************4&F0$\"3))* *************HF*7$FA$\"33+++++++EF0F1-F$6$7$7$$\"3M+++++++zF0FC7$FLFFF 1-F$6$7$7$$\"3k*************4*F0FC7$FSFFF1-F$6$7$7$$\"3/++++++!>\"!#;F C7$FZFFF1-%%TEXTG6&7$$F5F5$\"\"$F5Q\"06\"F1-%%FONTG6%%&TIMESG%&ROMANG \"#7-Fin6&7$$\"\"\"F5F]oQ\"1F`oF1Fao-Fin6&7$$\"\"#F5F]oQ\"4F`oF1Fao-Fi n6&7$F]oF]oQ\"5F`oF1Fao-Fin6&7$F\\oF`pQ\"3F`oF1Fao-Fin6&7$FjoF`pF\\pF1 Fao-Fin6&7$F`pF`pQ\"2F`oF1Fao-Fin6&7$F]oF`pF\\pF1Fao-Fin6&7$F\\oFjoF_o F1Fao-Fin6&7$FjoFjoF\\pF1Fao-Fin6&7$F`pFjoFjpF1Fao-Fin6&7$F]oFjoF_oF1F ao-Fin6&7$F\\oF\\oFaqF1Fao-Fin6&7$FjoF\\oF\\pF1Fao-Fin6&7$F`pF\\oF_oF1 Fao-Fin6&7$F]oF\\oFjpF1Fao-Fin6&7$$\"#b!\"\"$\"#DFbsF\\pF1Fao-Fin6&7$$ \"#lFbsFcsFbpF1Fao-Fin6&7$$\"#vFbsFcsFfpF1Fao-Fin6&7$F`s$\"#:FbsF\\pF1 Fao-Fin6&7$FhsFbtFjpF1Fao-Fin6&7$F]tFbtF_oF1Fao-Fin6&7$F`s$\"\"&FbsF\\ pF1Fao-Fin6&7$FhsF]uF_oF1Fao-Fin6&7$F]tF]uFjpF1Fao-Fin6&7$$\"#&*FbsFcs F\\pF1Fao-Fin6&7$$\"$0\"FbsFcsFbpF1Fao-Fin6&7$$\"$:\"FbsFcsFfpF1Fao-Fi n6&7$FhuFbtF\\pF1Fao-Fin6&7$F]vFbtFaqF1Fao-Fin6&7$FbvFbtF\\pF1Fao-Fin6 &7$FhuF]uF\\pF1Fao-Fin6&7$F]vF]uFjpF1Fao-Fin6&7$FbvF]uF_oF1Fao-Fin6&7$ $\"#TFbs$\"$Y\"!\"#Q\"=F`oF1Fao-Fin6&7$$\"#ZFbsFbtQ$-~3F`oF1Fao-Fin6&7 $$\"#')FbsFbtQ$-~2F`oF1Fao-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F`oF by-%%VIEWG6$%(DEFAULTGFfy" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 274 166 166 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+ ++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F +7$F:F.F1-F$6(7$7$$!\"\"!\"#FA7$$\"$q#FCFE7%7$$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7 %7$$\"+\"*pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\" \"$FBF^rFYFgn-F$6(7$7$F]oF]p7$F`oF^q7%7$$\"+/gNFCFA7$$\"$q%FCFE7 %7$$\"+\"*pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\" \"$FBF^rFYFgn-F$6(7$7$F]oF]p7$F`oF^q7%7$$\"+/gN " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Properties of determinants " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 15 "" 0 "" {TEXT -1 84 " If two rows ( or columns ) are interchanged the sign of the determinan t is changed. " }}{PARA 15 "" 0 "" {TEXT -1 69 "If two rows ( or colum ns ) are identical, then the determinant is 0. " }}{PARA 15 "" 0 "" {TEXT -1 67 "If all entries in a row ( or column ) are multiplied by a constant " }{TEXT 277 1 "k" }{TEXT -1 35 ", the determinant is multip lied by " }{TEXT 278 1 "k" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 96 "If a multiple of one row ( or column ) is added to another row, the determinant is not changed. " }}{PARA 15 "" 0 "" {TEXT -1 3 "If \+ " }{TEXT 280 1 "A" }{TEXT -1 5 " and " }{TEXT 279 1 "B" }{TEXT -1 54 " are two square matrices of the same dimension, then " }{XPPEDIT 18 0 "det(A*`.`*B)=det(A)*`.`*det(B)" "6#/-%$detG6#*(%\"AG\"\"\"%\".GF)% \"BGF)*(-F%6#F(F)F*F)-F%6#F+F)" }{TEXT -1 3 ". \n" }}{PARA 15 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "I = matrix([[1, 0, 0, ` . . . `, 0], [0, 1, 0, ` . . . `, 0], [0, 0, 1, ` . . . `, 0], [``, ``, ``, ``, `` ], [0, 0, 0, ` . . . `, 1]]);" "6#/%\"IG-%'matrixG6#7'7'\"\"\"\"\"!F+% (~.~.~.~GF+7'F+F*F+F,F+7'F+F+F*F,F+7'%!GF0F0F0F07'F+F+F+F,F*" }{TEXT -1 30 " is an identity matrix, then " }{XPPEDIT 18 0 "det(I) = 1;" "6 #/-%$detG6#%\"IG\"\"\"" }{TEXT -1 3 ". \n" }}{PARA 15 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "A = matrix([[a[1,1], a[1,2], ` . . . `, a[1 ,n]], [0, a[2,2], ` . . . `, a[n,1]], [``, ``, ``, ``], [0, 0, ` . . . `, a[n.n]]])" "6#/%\"AG-%'matrixG6#7&7&&%\"aG6$\"\"\"F-&F+6$F-\"\"#%( ~.~.~.~G&F+6$F-%\"nG7&\"\"!&F+6$F0F0F1&F+6$F4F-7&%!GF " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 25 "The determinant of the 4 " }{TEXT 295 1 "x" }{TEXT -1 10 " 4 matrix " }{XPPEDIT 18 0 "A = matrix([[0, 1, 4, 5], [3, 1, 2, 1], [ 0, 1, 3, 0], [2, 1, 0, 3]])" "6#/%\"AG-%'matrixG6#7&7&\"\"!\"\"\"\"\"% \"\"&7&\"\"$F+\"\"#F+7&F*F+F/F*7&F0F+F*F/" }{TEXT -1 80 " considered e arlier can be evaluated with the aid of row operations as follows. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Subtrac ting the 4th row from the 2nd row and then subtracting twice the 2nd r ow from the 4th row gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 566 152 152 {PLOTDATA 2 "6gn-%'CURVESG6$7$7$$!3A+++++++S!#=$ !35+++++++?F*7$F($\"33+++++++J!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$ \"3!**************R$F0F+7$F:F.F1-F$6$7$7$$\"3k*************f%F0F+7$FAF .F1-F$6$7$7$$\"3M+++++++%)F0F+7$FHF.F1-F$6$7$7$$\"3k*************f*F0F +7$FOF.F1-F$6$7$7$$\"3/++++++S8!#;F+7$FVF.F1-%%TEXTG6&7$$F5F5$\"\"$F5Q \"06\"F1-%%FONTG6%%&TIMESG%&ROMANG\"#7-Fen6&7$$\"\"\"F5FinQ\"1F\\oF1F] o-Fen6&7$$\"\"#F5FinQ\"4F\\oF1F]o-Fen6&7$FinFinQ\"5F\\oF1F]o-Fen6&7$Fh nF\\pQ\"3F\\oF1F]o-Fen6&7$FfoF\\pFhoF1F]o-Fen6&7$F\\pF\\pQ\"2F\\oF1F]o -Fen6&7$FinF\\pFhoF1F]o-Fen6&7$FhnFfoF[oF1F]o-Fen6&7$FfoFfoFhoF1F]o-Fe n6&7$F\\pFfoFfpF1F]o-Fen6&7$FinFfoF[oF1F]o-Fen6&7$FhnFhnF]qF1F]o-Fen6& 7$FfoFhnFhoF1F]o-Fen6&7$F\\pFhnF[oF1F]o-Fen6&7$FinFhnFfpF1F]o-Fen6&7$$ \"\"&F5FinF[oF1F]o-Fen6&7$$\"\"'F5FinFhoF1F]o-Fen6&7$$\"\"(F5FinF^pF1F ]o-Fen6&7$$\"\")F5FinFbpF1F]o-Fen6&7$F\\sF\\pFhoF1F]o-Fen6&7$FasF\\pF[ oF1F]o-Fen6&7$FfsF\\pF]qF1F]o-Fen6&7$F[tF\\pQ#-2F\\oF1F]o-Fen6&7$F\\sF foF[oF1F]o-Fen6&7$FasFfoFhoF1F]o-Fen6&7$FfsFfoFfpF1F]o-Fen6&7$F[tFfoF[ oF1F]o-Fen6&7$F\\sFhnF]qF1F]o-Fen6&7$FasFhnFhoF1F]o-Fen6&7$FfsFhnF[oF1 F]o-Fen6&7$F[tFhnFfpF1F]o-Fen6&7$$\"#5F5FinF[oF1F]o-Fen6&7$$\"#6F5FinF hoF1F]o-Fen6&7$$FboF5FinF^pF1F]o-Fen6&7$$\"#8F5FinFbpF1F]o-Fen6&7$FevF \\pFhoF1F]o-Fen6&7$FjvF\\pF[oF1F]o-Fen6&7$F_wF\\pF]qF1F]o-Fen6&7$FcwF \\pFitF1F]o-Fen6&7$FevFfoF[oF1F]o-Fen6&7$FjvFfoFhoF1F]o-Fen6&7$F_wFfoF fpF1F]o-Fen6&7$FcwFfoF[oF1F]o-Fen6&7$FevFhnF[oF1F]o-Fen6&7$FjvFhnFhoF1 F]o-Fen6&7$F_wFhnQ#-4F\\oF1F]o-Fen6&7$FcwFhnQ\"7F\\oF1F]o-Fen6&7$$\"\" %F5$\"#:!\"\"Q\"=F\\oF1F]o-Fen6&7$$\"\"*F5F`zFczF1F]o-%*AXESSTYLEG6#%% NONEG-%+AXESLABELSG6$Q!F\\oF`[l-%%VIEWG6$%(DEFAULTGFd[l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 47.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Cur ve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 2 9" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" " Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curv e 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48 " "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "C urve 55" "Curve 56" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 103 "T hen expanding the last determinant down the 1st column and subtracting the 1st row of the resulting 3 " }{TEXT 294 1 "x" }{TEXT -1 43 " 3 ma trix from the 2nd and 3rd rows gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 430 125 125 {PLOTDATA 2 "6D-%'CURVESG6$7$7$$!3A++++++ +S!#=$!35+++++++?F*7$F($\"33+++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6 $7$7$$\"3!**************R#F0F+7$F:F.F1-F$6$7$7$$\"33+++++++OF0F+7$FAF. F1-F$6$7$7$$\"3M+++++++kF0F+7$FHF.F1-F$6$7$7$$\"3k*************f(F0$\" 3))**************HF*7$FO$\"33+++++++;F0F1-F$6$7$7$$\"3M+++++++%*F0FQ7$ FZFTF1-%%TEXTG6&7$$F5F5$\"\"#F5Q\"16\"F1-%%FONTG6%%&TIMESG%&ROMANG\"#7 -Fhn6&7$$\"\"\"F5F\\oQ\"4F_oF1F`o-Fhn6&7$F\\oF\\oQ\"5F_oF1F`o-Fhn6&7$F [oFioF^oF1F`o-Fhn6&7$FioFioQ\"3F_oF1F`o-Fhn6&7$F\\oFioQ\"0F_oF1F`o-Fhn 6&7$F[oF[oF^oF1F`o-Fhn6&7$FioF[oQ#-4F_oF1F`o-Fhn6&7$F\\oF[oQ\"7F_oF1F` o-Fhn6&7$$\"\"%F5F\\oF^oF1F`o-Fhn6&7$$\"\"&F5F\\oF[pF1F`o-Fhn6&7$$\"\" 'F5F\\oF_pF1F`o-Fhn6&7$FiqFioFjpF1F`o-Fhn6&7$F^rFioQ#-1F_oF1F`o-Fhn6&7 $FcrFioQ#-5F_oF1F`o-Fhn6&7$FiqF[oFjpF1F`o-Fhn6&7$F^rF[oQ#-8F_oF1F`o-Fh n6&7$FcrF[oQ\"2F_oF1F`o-Fhn6&7$$\"\")F5$\"#:!\"\"F[sF1F`o-Fhn6&7$$\"\" *F5F`tF_sF1F`o-Fhn6&7$F^t$F_rFbtFfsF1F`o-Fhn6&7$FftF[uFjsF1F`o-Fhn6&7$ $!\"(FbtFioQ\"-F_oF1F`o-Fhn6&7$$\"#JFbtFioQ%=~~-F_oF1F`o-Fhn6&7$$\"#rF btFioFjuF1F`o-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F_oFgv-%%VIEWG6$% (DEFAULTGF[w" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20 " "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "C urve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" }}{TEXT -1 2 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -(-2-40) " " 6#/%!G,$,&\"\"#!\"\"\"#SF(F(" }{XPPEDIT 18 0 "``=42" "6#/%!G\"#U" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 281 1 "A" }{TEXT -1 23 " has an inverse matrix " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 16 ", then we have: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*`.`*A^(-1)=I" "6#/* (%\"AG\"\"\"%\".GF&)F%,$F&!\"\"F&%\"IG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "det(A*`.`*A^(-1))=det(I)" "6#/-%$detG6#*(%\"AG\"\"\"%\".GF))F(,$ F)!\"\"F)-F%6#%\"IG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "det(A)*`.`* det(A^(-1))=1" "6#/*(-%$detG6#%\"AG\"\"\"%\".GF)-F&6#)F(,$F)!\"\"F)F) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "det(A^(-1)) = det(A)^ (-1);" "6#/-%$detG6#)%\"AG,$\"\"\"!\"\")-F%6#F(,$F*F+" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 13 "_____________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the relation " }{XPPEDIT 18 0 "det(A)*`.`*det(A ^(-1)) = 1" "6#/*(-%$detG6#%\"AG\"\"\"%\".GF)-F&6#)F(,$F)!\"\"F)F)" } {TEXT -1 28 " means, in particular, that " }{XPPEDIT 18 0 "det(A)<>0" "6#0-%$detG6#%\"AG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "Thus if a matrix A has an inverse " }{XPPEDIT 18 0 "A^(-1)" "6#)% \"AG,$\"\"\"!\"\"" }{TEXT -1 38 ", that is, if A is non-singular, then " }{XPPEDIT 18 0 "det(A)<>0" "6#0-%$detG6#%\"AG\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 "The converse of the last statement al so holds. " }}{PARA 0 "" 0 "" {TEXT 259 9 "A matrix " }{TEXT 283 1 "A " }{TEXT 259 52 " has an inverse (is non-singular) exactly when det( \+ " }{TEXT 284 2 "A " }{TEXT 259 1 ")" }{XPPEDIT 18 0 "`` <> 0;" "6#0%!G \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "A=matrix([[3,1],[-1,2 ]])" "6#/%\"AG-%'matrixG6#7$7$\"\"$\"\"\"7$,$F+!\"\"\"\"#" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "det(A)=7" "6#/-%$detG6#%\"AG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A^(-1)=1/7" "6#/)%\"AG,$\"\"\"!\"\"*&F' F'\"\"(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2,-1],[1,3]]) =mat rix([[2/7,-1/7],[1/7,3/7]])" "6#/-%'matrixG6#7$7$\"\"#,$\"\"\"!\"\"7$F +\"\"$-F%6#7$7$*&F)F+\"\"(F,,$*&F+F+F4F,F,7$*&F+F+F4F,*&F.F+F4F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "det (A^(-1))=7/49" "6#/-%$detG6#)%\"AG,$\"\"\"!\"\"*&\"\"(F*\"#\\F+" } {XPPEDIT 18 0 "``=1/7" "6#/%!G*&\"\"\"F&\"\"(!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "F or another example, let " }{XPPEDIT 18 0 "A=matrix([[1,2,-1],[2,1,5],[ 3,4,1]])" "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#,$F*!\"\"7%F+F*\"\"&7%\" \"$\"\"%F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "det(A) = ``;" "6#/-%$ detG6#%\"AG%!G" }{TEXT -1 5 " det " }{XPPEDIT 18 0 "matrix([[1, 2, -1] , [2, 1, 5], [3, 4, 1]]) = ``;" "6#/-%'matrixG6#7%7%\"\"\"\"\"#,$F)!\" \"7%F*F)\"\"&7%\"\"$\"\"%F)%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "mat rix([[1, 2, -1], [0, -3, 7], [0, -2, 4]]);" "6#-%'matrixG6#7%7%\"\"\" \"\"#,$F(!\"\"7%\"\"!,$\"\"$F+\"\"(7%F-,$F)F+\"\"%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= -``" "6#/%!G,$F$!\"\"" }{TEXT -1 4 "det " } {XPPEDIT 18 0 "matrix([[1, 2, -1], [0, -2, 4], [0, -3, 7]]) = 2;" "6#/ -%'matrixG6#7%7%\"\"\"\"\"#,$F)!\"\"7%\"\"!,$F*F,\"\"%7%F.,$\"\"$F,\" \"(F*" }{TEXT -1 5 " det " }{XPPEDIT 18 0 "matrix([[1, 2, -1], [0, 1, \+ -2], [0, -3, 7]]) = 2;" "6#/-%'matrixG6#7%7%\"\"\"\"\"#,$F)!\"\"7%\"\" !F),$F*F,7%F.,$\"\"$F,\"\"(F*" }{TEXT -1 5 " det " }{XPPEDIT 18 0 "mat rix([[1, 2, -1], [0, 1, -2], [0, 0, 1]]) = 2;" "6#/-%'matrixG6#7%7%\" \"\"\"\"#,$F)!\"\"7%\"\"!F),$F*F,7%F.F.F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We now use Gaus s-Jordan elimination to find " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, 1, 0, 0], [2,1,5, `|`, 0, 1, 0] , [3,4,1, `|`, 0, 0, 1]])" "6#-%'matrixG6#7%7)\"\"\"\"\"#,$F(!\"\"%\"| grGF(\"\"!F-7)F)F(\"\"&F,F-F(F-7)\"\"$\"\"%F(F,F-F-F(" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-2*R[1], ``],[R[3] = R[3]-3*R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&F+F/&F)6#F/ F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F9F/*&F9F/&F)6#F/F/F3F4" }{TEXT -1 6 " \+ ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, 1, 0, 0], [0, -3,7, `|`, -2, 1, 0], [0, -2,4, `|`, -3, 0, 1]])" "6#-%'matrixG6#7%7)\"\"\"\"\"# ,$F(!\"\"%\"|grGF(\"\"!F-7)F-,$\"\"$F+\"\"(F,,$F)F+F(F-7)F-,$F)F+\"\"% F,,$F0F+F-F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " swap " }{XPPEDIT 18 0 "R[2]" "6#&%\"RG6#\"\"# " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R[3]" "6#&%\"RG6#\"\"$" }{TEXT -1 6 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, 1, 0, 0],[0, -2, 4, `|`, -3, 0, 1], [0, -3, 7, `|`, -2, 1, 0]])" "6#-%'matrixG6#7%7)\" \"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F-7)F-,$F)F+\"\"%F,,$\"\"$F+F-F(7)F-,$ F2F+\"\"(F,,$F)F+F(F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2] = R[2]*`.`*` `(-1/2);" "6#/&%\"RG6#\"\"#*(&F%6#F'\"\"\"%\".GF+-%!G6#,$*&F+F+F'!\"\" F2F+" }{TEXT -1 6 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, 1, \+ 0, 0], [0, 1, -2, `|`, 3/2, 0, -1/2], [0, -3, 7, `|`, -2, 1, 0]]);" "6 #-%'matrixG6#7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F-7)F-F(,$F)F+F,*& \"\"$F(F)F+F-,$*&F(F(F)F+F+7)F-,$F1F+\"\"(F,,$F)F+F(F-" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R[3] = R[3]+3*R[2];" "6#/&%\"RG6#\"\"$,&&F%6#F'\"\"\"*& F'F+&F%6#\"\"#F+F+" }{TEXT -1 6 " ... " }{XPPEDIT 18 0 "matrix([[1, 2 , -1, `|`, 1, 0, 0], [0, 1, -2, `|`, 3/2, 0, -1/2], [0, 0, 1, `|`, 5/2 , 1, -3/2]]);" "6#-%'matrixG6#7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F- 7)F-F(,$F)F+F,*&\"\"$F(F)F+F-,$*&F(F(F)F+F+7)F-F-F(F,*&\"\"&F(F)F+F(,$ *&F1F(F)F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[1] = R[1]+R[3] , ``],[R[2] = R[2]+2*R[1], ``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\",& &F)6#F+F+&F)6#\"\"$F+%!G7$/&F)6#\"\"#,&&F)6#F7F+*&F7F+&F)6#F+F+F+F2" } {TEXT -1 6 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, 0, `|`, 7/2, 1, -3/ 2], [0, 1, 0, `|`, 13/2, 2, -7/2], [0, 0, 1, `|`, 5/2, 1, -3/2]]);" "6 #-%'matrixG6#7%7)\"\"\"\"\"#\"\"!%\"|grG*&\"\"(F(F)!\"\"F(,$*&\"\"$F(F )F.F.7)F*F(F*F+*&\"#8F(F)F.F),$*&F-F(F)F.F.7)F*F*F(F+*&\"\"&F(F)F.F(,$ *&F1F(F)F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1] = R[1]-2*R[2];" "6#/&% \"RG6#\"\"\",&&F%6#F'F'*&\"\"#F'&F%6#F,F'!\"\"" }{TEXT -1 6 " ... " } {XPPEDIT 18 0 "matrix([[1, 2, 0, `|`, (-19)/2, -3, 11/2], [0, 1, 0, `| `, 13/2, 2, -7/2], [0, 0, 1, `|`, 5/2, 1, -3/2]]);" "6#-%'matrixG6#7%7 )\"\"\"\"\"#\"\"!%\"|grG*&,$\"#>!\"\"F(F)F/,$\"\"$F/*&\"#6F(F)F/7)F*F( F*F+*&\"#8F(F)F/F),$*&\"\"(F(F)F/F/7)F*F*F(F+*&\"\"&F(F)F/F(,$*&F1F(F) F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "A^(-1)=matrix([[-19/2, -3, 11/2] , [13/2, 2, -7/2], [5/2, 1, -3/2]])" "6#/)%\"AG,$\"\"\"!\"\"-%'matrixG 6#7%7%,$*&\"#>F'\"\"#F(F(,$\"\"$F(*&\"#6F'F1F(7%*&\"#8F'F1F(F1,$*&\"\" (F'F1F(F(7%*&\"\"&F'F1F(F',$*&F3F'F1F(F(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " det(A^(-1))=1/8" "6#/-%$detG6#)%\"AG,$\"\"\"!\"\"*&F*F*\"\")F+" } {TEXT -1 5 " det " }{XPPEDIT 18 0 "matrix([[-19, -6, 11], [13, 4, -7], [5, 2, -3]])" "6#-%'matrixG6#7%7%,$\"#>!\"\",$\"\"'F*\"#67%\"#8\"\"%, $\"\"(F*7%\"\"&\"\"#,$\"\"$F*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 365 228 228 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$ !3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++@!#<-%'COLOURG6&%$RGBG\"\" !F5F5-F$6$7$7$$\"3!**************R#F0F+7$F:F.F1-F$6(7$7$$!\"\"!\"#FA7$ $\"$q#FCFE7%7$$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7%7$$\"+\"*pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$ \"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\"\"$FBF^rFYFgn-F$6(7$7$F]oF]p7$F`oF^ q7%7$$\"+/gN " 0 "" {MPLTEXT 1 0 110 "A := Matrix([[1,2,-1],[ 2,1,5],[3,4,1]]);\nLinearAlgebra[Determinant](A);\nA^(-1);\nLinearAlge bra[Determinant](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABL EG6%\"*[@6X\"-%'MATRIXG6#7%7%\"\"\"\"\"#!\"\"7%F/F.\"\"&7%\"\"$\"\"%F. %'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6%\"*;l6X\"-%'MATRIXG6#7%7%#!#>\"\"#!\"$#\" #6F.7%#\"#8F.F.#!\"(F.7%#\"\"&F.\"\"\"#F/F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Evaluate \+ the determinant of each of the following matrices. " }}{PARA 0 "" 0 " " {TEXT -1 12 " (a) " }{XPPEDIT 18 0 "matrix([[1,1,5],[4,3,6],[ 0,-1,1]])" "6#-%'matrixG6#7%7%\"\"\"F(\"\"&7%\"\"%\"\"$\"\"'7%\"\"!,$F (!\"\"F(" }{TEXT -1 29 " (b) " }{XPPEDIT 18 0 "matrix([[2,4, 5], [4, 2, 0], [8,7,-2]])" "6#-%'matrixG6#7%7%\"\"#\"\" %\"\"&7%F)F(\"\"!7%\"\")\"\"(,$F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " (c) " } {XPPEDIT 18 0 "matrix([[-1,2,3], [4, 5,-2], [3,-3,2]])" "6#-%'matrixG6 #7%7%,$\"\"\"!\"\"\"\"#\"\"$7%\"\"%\"\"&,$F+F*7%F,,$F,F*F+" }{TEXT -1 27 " (d) " }{XPPEDIT 18 0 "matrix([[1,2,3,4], [1 ,3,5,7], [2,3,6,7],[1,5,8,20]])" "6#-%'matrixG6#7&7&\"\"\"\"\"#\"\"$\" \"%7&F(F*\"\"&\"\"(7&F)F*\"\"'F.7&F(F-\"\")\"#?" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) -15 (b) 84 (c) -113 (d) 1 6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "A := Matrix([[1, 1, 5], [4, 3, 6], [0, -1, 1]]);\nLi nearAlgebra[Determinant](A);\nA := Matrix([[2,4, 5], [4, 2, 0], [8,7,- 2]]);\nLinearAlgebra[Determinant](A);\nA := Matrix([[-1,2,3], [4, 5,-2 ], [3,-3,2]]);\nLinearAlgebra[Determinant](A);\nA := Matrix([[1,2,3,4] , [1,3,5,7], [2,3,6,7],[1,5,8,20]]);\nLinearAlgebra[Determinant](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*k#>^9-%'MATRIXG 6#7%7%\"\"\"F.\"\"&7%\"\"%\"\"$\"\"'7%\"\"!!\"\"F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A G-%'RTABLEG6%\"*3L7X\"-%'MATRIXG6#7%7%\"\"#\"\"%\"\"&7%F/F.\"\"!7%\"\" )\"\"(!\"#%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*%3D^9-%'MATRIXG6#7%7%! \"\"\"\"#\"\"$7%\"\"%\"\"&!\"#7%F0!\"$F/%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$8\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABL EG6%\"*Ol7X\"-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&F.F0\"\"&\"\"(7&F/ F0\"\"'F47&F.F3\"\")\"#?%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "__________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 17 "For the matr ices " }{XPPEDIT 18 0 "A=matrix([[2,-1,1],[3,1,-1],[0,2,2]])" "6#/%\"A G-%'matrixG6#7%7%\"\"#,$\"\"\"!\"\"F,7%\"\"$F,,$F,F-7%\"\"!F*F*" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B=matrix([[2,1,5],[4,3,8],[0,-1,0]] )" "6#/%\"BG-%'matrixG6#7%7%\"\"#\"\"\"\"\"&7%\"\"%\"\"$\"\")7%\"\"!,$ F+!\"\"F2" }{TEXT -1 14 ", verify that " }{XPPEDIT 18 0 "det(A*`.`*B)= det(A)*det(B)" "6#/-%$detG6#*(%\"AG\"\"\"%\".GF)%\"BGF)*&-F%6#F(F)-F%6 #F+F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "det(A)=20" "6#/-%$detG 6#%\"AG\"#?" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "det(B)=-4" "6#/-%$detG6# %\"BG,$\"\"%!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "A*`.`*B=matrix([ [0, -2, 2], [10, 7, 23], [8, 4, 16]])" "6#/*(%\"AG\"\"\"%\".GF&%\"BGF& -%'matrixG6#7%7%\"\"!,$\"\"#!\"\"F07%\"#5\"\"(\"#B7%\"\")\"\"%\"#;" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "det(A*`.`*B)=-80" "6#/-%$detG6#*(% \"AG\"\"\"%\".GF)%\"BGF),$\"#!)!\"\"" }{XPPEDIT 18 0 "``= det(A)*det(B )" "6#/%!G*&-%$detG6#%\"AG\"\"\"-F'6#%\"BGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "A := Matrix([[2,-1,1],[3,1,-1],[0,2,2]]);\nB := Matrix([[2, 1, 5], [4, \+ 3, 8], [0, -1, 0]]);\nA.B:\nconvert(%,matrix);\nLinearAlgebra[Determin ant](A);\nLinearAlgebra[Determinant](B);\nLinearAlgebra[Determinant](A .B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*K48X\"-%' MATRIXG6#7%7%\"\"#!\"\"\"\"\"7%\"\"$F0F/7%\"\"!F.F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"*!oL^9-%'MATRIXG6#7%7%\" \"#\"\"\"\"\"&7%\"\"%\"\"$\"\")7%\"\"!!\"\"F6%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!!\"#\"\"#7%\"#5\"\"(\"#B7%\" \")\"\"%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#!)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 15 "For the matrix " } {XPPEDIT 18 0 "A = matrix([[1,0,5],[1,1,2],[2,1,-1]])" "6#/%\"AG-%'mat rixG6#7%7%\"\"\"\"\"!\"\"&7%F*F*\"\"#7%F.F*,$F*!\"\"" }{TEXT -1 14 ", \+ verify that " }{XPPEDIT 18 0 "det(A^(-1)) = det(A)^(-1);" "6#/-%$detG6 #)%\"AG,$\"\"\"!\"\")-F%6#F(,$F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "det(A)=-8" "6#/-%$detG6#%\"AG,$ \"\")!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "A^(-1)=matrix([[3/8, -5 /8, 5/8], [-5/8, 11/8, -3/8], [1/8, 1/8, -1/8]])" "6#/)%\"AG,$\"\"\"! \"\"-%'matrixG6#7%7%*&\"\"$F'\"\")F(,$*&\"\"&F'F0F(F(*&F3F'F0F(7%,$*&F 3F'F0F(F(*&\"#6F'F0F(,$*&F/F'F0F(F(7%*&F'F'F0F(*&F'F'F0F(,$*&F'F'F0F(F (" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "det(A^(-1))=-1/8" "6#/-%$detG 6#)%\"AG,$\"\"\"!\"\",$*&F*F*\"\")F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "A := M atrix([[1,0,5],[1,1,2],[2,1,-1]]);\nA^(-1):\nconvert(%,matrix);\nLinea rAlgebra[Determinant](A);\nLinearAlgebra[Determinant](A^(-1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*K^8X\"-%'MATRIXG6 #7%7%\"\"\"\"\"!\"\"&7%F.F.\"\"#7%F2F.!\"\"%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"$\"\")#!\"&F*#\"\"&F*7%F+#\"#6F *#!\"$F*7%#\"\"\"F*F5#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pi ctures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "2 x 2 determinant" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "p1 := plot( [[[2.6,-.2],[2.6,1.2]],[[4.4,-.2],[4.4,1.2]]],color=black):\nt1 := plo ts[textplot]([[3,1,`a`],[4,1,`b`],[3,0,`c`],[4,0,`d`],[1.82,.53,`A`], \n [4.77,.5,`=`],[5.6,.53,`a d - b c`]],font=[TIMES,ITALIC,12],color=b lack):\nt2 := plots[textplot]([1.77,.53,`det( ) =`],font=[TIMES,ROMA N,12],color=black):\nplots[display]([p1,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 488 "p 1 := plot([[[4.1,.3],[4.1,1.6]],[[5.9,.3],[5.9,1.6]],\n [[7.1,.3],[ 7.1,1.6]],[[8.9,.3],[8.9,1.6]],\n [[10.1,.3],[10.1,1.6]],[[11.9,.3] ,[11.9,1.6]]],color=black):\nt1 := plots[textplot]([[4.5,1.5,`e`],[5.5 ,1.5,`f`],[4.5,.5,`h`],[5.5,.5,`k`],\n [7.5,1.5,`d`],[8.5,1.5,`f`],[7 .5,.5,`g`],[8.5,.5,`k`],\n [10.5,1.5,`d`],[11.5,1.5,`e`],[10.5,.5,`g` ],[11.5,.5,`h`],\n [3,.96,`=`],[3.6,1,`a`],[6.6,1,`- b`],[9.6,1,`+ c`] ],font=[TIMES,ITALIC,12],color=black):\nplots[display]([p1,t1],axes=no ne);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 44 "expansion \+ of 3 x 3 determinant along 1st row" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 735 "p1 := plot([[[-.4,-.2],[-.4 ,2.1]],[[2.4,-.2],[2.4,2.1]],\n [[4.1,.3],[4.1,1.6]],[[5.9,.3],[5.9 ,1.6]],\n [[7.1,.3],[7.1,1.6]],[[8.9,.3],[8.9,1.6]],\n [[10.1,.3 ],[10.1,1.6]],[[11.9,.3],[11.9,1.6]]],color=black):\nt1 := plots[textp lot]([[0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n \+ [0,0,`g`],[1,0,`h`],[2,0,`k`],\n [4.5,1.5,`e`],[5.5,1.5,`f`],[4.5,.5, `h`],[5.5,.5,`k`],\n [7.5,1.5,`d`],[8.5,1.5,`f`],[7.5,.5,`g`],[8.5,.5 ,`k`],\n [10.5,1.5,`d`],[11.5,1.5,`e`],[10.5,.5,`g`],[11.5,.5,`h`],\n [-1.42,1,`A`],\n [3.1,.96,`=`],[3.7,1,`a`],[6.6,1,`- b`],[9.6,1,`+ c `]],font=[TIMES,ITALIC,12],color=black):\nt2 := plots[textplot]([-1.5, 1,`det( ) =`],font=[TIMES,ROMAN,12],color=black):\nplots[display]([p 1,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 707 "p1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4, -.2],[2.4,2.1]],\n [[4.6,.3],[4.6,1.6]],[[6.4,.3],[6.4,1.6]]],color =black):\nt1 := plots[textplot]([[0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d `],[1,1,`e`],[2,1,`f`],\n [0,0,`g`],[1,0,`h`],[2,0,`k`],\n [5,1.5,`e `],[6,1.5,`f`],[5,.5,`h`],[6,.5,`k`],\n [4.2,1,`+ a`]],font=[TIMES,IT ALIC,12],color=black):\np2 := plottools[arrow]([2.7,1],[3.7,1],0,.13,. 17,arrow,color=red):\np3 := plots[polygonplot]([[[-.2,-.2],[-.2,2.1],[ .2,2.1],[.2,-.2]],\n [[-.2,2.1],[2.2,2.1],[2.2,1.8],[-.2,1.8]]],\n style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10) :\np4 := plot([seq([.2*cos(i*dd),1.95+.2*sin(i*dd)],i=0..20)],color=re d): \nplots[display]([p1,p2,p3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 709 "p1 := plot( [[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]],\n [[4.6,.3],[4.6,1.6] ],[[6.4,.3],[6.4,1.6]]],color=black):\nt1 := plots[textplot]([[0,2,`a` ],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n [0,0,`g`],[1,0 ,`h`],[2,0,`k`],\n [5,1.5,`d`],[6,1.5,`f`],[5,.5,`g`],[6,.5,`k`],\n \+ [4.2,1,`- b`]],font=[TIMES,ITALIC,12],color=black):\np2 := plottools[a rrow]([2.7,1],[3.7,1],0,.13,.17,arrow,color=red):\np3 := plots[polygon plot]([[[.8,-.2],[.8,2.1],[1.2,2.1],[1.2,-.2]],\n [[-.2,2.1],[2.2, 2.1],[2.2,1.8],[-.2,1.8]]],\n style=patchnogrid,color=COLOR(RGB,.9 ,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([1+.2*cos(i*dd),1.95+. 2*sin(i*dd)],i=0..20)],color=red): \nplots[display]([p1,p2,p3,p4,t1],a xes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 711 "p1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4, 2.1]],\n [[4.6,.3],[4.6,1.6]],[[6.4,.3],[6.4,1.6]]],color=black):\n t1 := plots[textplot]([[0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e `],[2,1,`f`],\n [0,0,`g`],[1,0,`h`],[2,0,`k`],\n [5,1.5,`d`],[6,1.5, `e`],[5,.5,`g`],[6,.5,`h`],\n [4.2,1,`+ c`]],font=[TIMES,ITALIC,12],c olor=black):\np2 := plottools[arrow]([2.7,1],[3.7,1],0,.13,.17,arrow,c olor=red):\np3 := plots[polygonplot]([[[1.8,-.2],[1.8,2.1],[2.2,2.1],[ 2.2,-.2]],\n [[-.2,2.1],[2.2,2.1],[2.2,1.8],[-.2,1.8]]],\n sty le=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([2+.2*cos(i*dd),1.95+.2*sin(i*dd)],i=0..20)],color=red): \n plots[display]([p1,p2,p3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "chequerboard of signs" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "p1 := plot([[[-.4,-.2],[-.4 ,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]([[0 ,2,`+`],[1,2,`-`],[2,2,`+`],[0,1,`-`],[1,1,`+`],[2,1,`-`],\n [0,0,`+` ],[1,0,`-`],[2,0,`+`]],font=[TIMES,BOLD,15],color=black):\np2 := plott ools[arrow]([2.7,1],[3.7,1],0,.13,.17,arrow,color=red): \nplots[displa y]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 46 "expansion of 3 x 3 determinant down 2nd column" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 709 "p 1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]],\n [[4.6,.3] ,[4.6,1.6]],[[6.4,.3],[6.4,1.6]]],color=black):\nt1 := plots[textplot] ([[0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n [0,0 ,`g`],[1,0,`h`],[2,0,`k`],\n [5,1.5,`d`],[6,1.5,`f`],[5,.5,`g`],[6,.5 ,`k`],\n [4.2,1,`- b`]],font=[TIMES,ITALIC,12],color=black):\np2 := p lottools[arrow]([2.7,1],[3.7,1],0,.13,.17,arrow,color=red):\np3 := plo ts[polygonplot]([[[.8,-.2],[.8,2.1],[1.2,2.1],[1.2,-.2]],\n [[-.2, 2.1],[2.2,2.1],[2.2,1.8],[-.2,1.8]]],\n style=patchnogrid,color=CO LOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([1+.2*cos(i* dd),1.95+.2*sin(i*dd)],i=0..20)],color=red): \nplots[display]([p1,p2,p 3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 706 "p1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4, -.2],[2.4,2.1]],\n [[4.6,.3],[4.6,1.6]],[[6.4,.3],[6.4,1.6]]],color =black):\nt1 := plots[textplot]([[0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d `],[1,1,`e`],[2,1,`f`],\n [0,0,`g`],[1,0,`h`],[2,0,`k`],\n [5,1.5,`a `],[6,1.5,`c`],[5,.5,`g`],[6,.5,`k`],\n [4.2,1,`+ e`]],font=[TIMES,IT ALIC,12],color=black):\np2 := plottools[arrow]([2.7,1],[3.7,1],0,.13,. 17,arrow,color=red):\np3 := plots[polygonplot]([[[.8,-.2],[.8,2.1],[1. 2,2.1],[1.2,-.2]],\n [[-.2,1.1],[2.2,1.1],[2.2,.8],[-.2,.8]]],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10): \np4 := plot([seq([1+.2*cos(i*dd),.95+.2*sin(i*dd)],i=0..20)],color=re d): \nplots[display]([p1,p2,p3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 707 "p1 := plot( [[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]],\n [[4.6,.3],[4.6,1.6] ],[[6.4,.3],[6.4,1.6]]],color=black):\nt1 := plots[textplot]([[0,2,`a` ],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n [0,0,`g`],[1,0 ,`h`],[2,0,`k`],\n [5,1.5,`a`],[6,1.5,`c`],[5,.5,`d`],[6,.5,`f`],\n \+ [4.2,1,`- h`]],font=[TIMES,ITALIC,12],color=black):\np2 := plottools[a rrow]([2.7,1],[3.7,1],0,.13,.17,arrow,color=red):\np3 := plots[polygon plot]([[[.8,-.2],[.8,2.1],[1.2,2.1],[1.2,-.2]],\n [[-.2,.1],[2.2,. 1],[2.2,-.2],[-.2,-.2]]],\n style=patchnogrid,color=COLOR(RGB,.9,. 9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([1+.2*cos(i*dd),-.05+.2* sin(i*dd)],i=0..20)],color=red): \nplots[display]([p1,p2,p3,p4,t1],axe s=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "p1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2. 1]],\n [[4.1,.3],[4.1,1.6]],[[5.9,.3],[5.9,1.6]],\n [[7.1,.3],[7 .1,1.6]],[[8.9,.3],[8.9,1.6]],\n [[10.1,.3],[10.1,1.6]],[[11.9,.3], [11.9,1.6]]],color=black):\nt1 := plots[textplot]([[0,2,`a`],[1,2,`b`] ,[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n [0,0,`g`],[1,0,`h`],[2,0, `k`],\n [4.5,1.5,`d`],[5.5,1.5,`f`],[4.5,.5,`g`],[5.5,.5,`k`],\n [7. 5,1.5,`a`],[8.5,1.5,`c`],[7.5,.5,`g`],[8.5,.5,`k`],\n [10.5,1.5,`a`], [11.5,1.5,`c`],[10.5,.5,`d`],[11.5,.5,`f`],\n [-1.42,1,`A`],\n [3.05, .96,`=`],[3.65,1,`- b`],[6.6,1,`+ e`],[9.6,1,`- h`]],font=[TIMES,ITALI C,12],color=black):\nt2 := plots[textplot]([-1.5,1,`det( ) =`],font= [TIMES,ROMAN,12],color=black):\nplots[display]([p1,t1,t2],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "direct exp ansion of 3 x 3 determinant " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1373 "p1 := plot([[[-.4,-.2],[-. 4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]([[ 0,2,`a`],[1,2,`b`],[2,2,`c`],[0,1,`d`],[1,1,`e`],[2,1,`f`],\n [0,0,`g `],[1,0,`h`],[2,0,`k`]],font=[TIMES,ITALIC,12],color=black):\nt2 := pl ots[textplot]([[3,2,`a`],[4,2,`b`],[3,1,`d`],[4,1,`e`],\n [3,0,`g`],[4 ,0,`h`]],font=[TIMES,ITALIC,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots [textplot]([[3,3,`g e c`],[4,3,`h f a`],[5,3,`k d b`]],\n font=[TIMES ,ITALIC,12],color=COLOR(RGB,0,0,.9)):\nt4 := plots[textplot]([[3,-1,`a e k`],[4,-1,`b f g`],[5,-1,`c d h`]],\n font=[TIMES,ITALIC,12],color =COLOR(RGB,.95,0,0)):\np2 := plottools[arrow]([-.01,-.01],[2.7,2.7],0, .15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2) :\np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arrow,\n \+ color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np4 := plottools[ar row]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,. 6,1),thickness=2,linestyle=2):\np5 := plottools[arrow]([-.01,1.99],[2. 7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,l inestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702],0,.15,.05, arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color =COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[display]([p1,p2,p 3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "p1 := plot([[[-.4,-.2] ,[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot ]([[0,2,`+`],[1,2,`+`],[2,2,`+`],[0,1,`+`],[1,1,`+`],[2,1,`+`],\n [0, 0,`+`],[1,0,`+`],[2,0,`+`]],font=[TIMES,BOLD,15],color=black):\np2 := \+ plottools[arrow]([2.7,1],[3.7,1],0,.13,.17,arrow,color=red): \nplots[d isplay]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 40 "numerical examples of 3 x 3 determinants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "TEMPLATE for expansi on along 1st row " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 852 "A := matrix([[a,b,c],[d,e,f],[g,h,k]]);\np1 \+ := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]],\n [[4.1,.3],[ 4.1,1.6]],[[5.9,.3],[5.9,1.6]],\n [[7.1,.3],[7.1,1.6]],[[8.9,.3],[8 .9,1.6]],\n [[10.1,.3],[10.1,1.6]],[[11.9,.3],[11.9,1.6]]],color=bl ack):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]], \n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3, 2]],[2,0,A[3,3]],\n [4.5,1.5,A[2,2]],[5.5,1.5,A[2,3]],[4.5,.5,A[3,2]] ,[5.5,.5,A[3,3]],\n [7.5,1.5,A[2,1]],[8.5,1.5,A[2,3]],[7.5,.5,A[3,1]] ,[8.5,.5,A[3,3]],\n [10.5,1.5,A[2,1]],[11.5,1.5,A[2,2]],[10.5,.5,A[3, 1]],[11.5,.5,A[3,2]],\n [3.7,1,A[1,1]],[6.7,1,A[1,2]],[9.7,1, A[1,3]] ],\n font=[TIMES,ITALIC,12],color=black):\nt2 := plots[textplot]([[ 3.1,.96,`=`],[6.25,.96,`-`],[9.25,.96,`+`]],\n font=[TIMES,ROMAN,12 ],color=black):\nplots[display]([p1,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "TEMPLATE for diagona l expansion " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1548 "A := matrix([[a,b,c],[d,e,f],[g,h,k]]);\np1 := p lot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2, 1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3 ]]],font=[TIMES,ITALIC,12],color=black):\nt2 := plots[textplot]([[3,2, A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[ 3,2]]],font=[TIMES,ITALIC,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[t extplot]([[3,3,-A[3,1]*A[2,2]*A[1,3]],[4,3,-A[3,2]*A[2,3]*A[1,1]],\n \+ [5,3,-A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ITALIC,12],color=COLOR(R GB,.3,0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1 ,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIME S,ITALIC,12],color=COLOR(RGB,1,0,.3)):\np2 := plottools[arrow]([-.01,- .01],[2.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickne ss=2,linestyle=2):\np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15, .05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np 4 := plottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n col or=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plottools[arrow ]([-.01,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3, .3),thickness=2,linestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7, -.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,lin estyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,a rrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[ display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 854 "A := matrix([[3,5,-6],[2,4 ,7],[8,-2,9]]);\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1 ]],\n [[4.1,.3],[4.1,1.6]],[[5.9,.3],[5.9,1.6]],\n [[7.1,.3],[7. 1,1.6]],[[8.9,.3],[8.9,1.6]],\n [[10.1,.3],[10.1,1.6]],[[11.9,.3],[ 11.9,1.6]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[ 1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0, 0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]],\n [4.5,1.5,A[2,2]],[5.5,1.5,A[2, 3]],[4.5,.5,A[3,2]],[5.5,.5,A[3,3]],\n [7.5,1.5,A[2,1]],[8.5,1.5,A[2, 3]],[7.5,.5,A[3,1]],[8.5,.5,A[3,3]],\n [10.5,1.5,A[2,1]],[11.5,1.5,A[ 2,2]],[10.5,.5,A[3,1]],[11.5,.5,A[3,2]],\n [3.7,1,A[1,1]],[6.7,1,A[1, 2]],[9.7,1, -A[1,3]]],\n font=[TIMES,ROMAN,12],color=black):\nt2 := plots[textplot]([[3.1,.96,`=`],[6.25,.96,`-`],[9.25,.96,`-`]],\n f ont=[TIMES,ROMAN,12],color=black):\nplots[display]([p1,t1,t2],axes=non e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1544 "A := matrix([[3,5,-6],[2,4,7],[8,-2,9]]);\np1 := pl ot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := \+ plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1 ]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3] ]],font=[TIMES,ROMAN,12],color=black):\nt2 := plots[textplot]([[3,2,A[ 1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[3, 2]]],font=[TIMES,ROMAN,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[text plot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3]*A[1,1]],\n [5,3, A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,0,0, .9)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2] *A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ROMAN ,12],color=COLOR(RGB,.95,0,0)):\np2 := plottools[arrow]([-.01,-.01],[2 .7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,li nestyle=2):\np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arr ow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np4 := pl ottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n color=COLO R(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plottools[arrow]([-.01 ,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thi ckness=2,linestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702], 0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle= 2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[display ]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1557 "A := matri x([[-19, -6, 11], [13, 4, -7], [5, 2, -3]]);\np1 := plot([[[-.4,-.2],[ -.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]( [[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]], [2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],font=[TIMES,R OMAN,12],color=black):\nt2 := plots[textplot]([[3,2,A[1,1]],[4,2,A[1,2 ]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[3,2]]],font=[TIMES ,ROMAN,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[textplot]([[3,3,A[3, 1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3]*A[1,1]],\n [5,3,A[3,3]* A[2,1]*A [1,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,0,0,.9)):\nt4 := plo ts[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2]*A[2,3]*A[3,1]], \n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ROMAN,12],color=COLOR (RGB,.95,0,0)):\np2 := plottools[arrow]([-.01,-.01],[2.7,2.7],0,.15,.0 5,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np3 \+ := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arrow,\n color= COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np4 := plottools[arrow]([ 1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),t hickness=2,linestyle=2):\np5 := plottools[arrow]([-.01,1.99],[2.7,-.70 2],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linesty le=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702],0,.15,.05,arrow, \n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np7 := plott ools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color=COLOR (RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[display]([p1,p2,p3,p4,p 5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 44 "expansion of 4 x 4 determinant along 1st row" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1214 " A := 'A':\np1 := plot([[[-.4,-.2],[-.4,3.1]],[[3.4,-.2],[3.4,3.1]],\n \+ [[5.1,.3],[5.1,2.6]],[[7.9,.3],[7.9,2.6]],\n [[9.1,.3],[9.1,2.6] ],[[11.9,.3],[11.9,2.6]],\n [[13.1,.3],[13.1,2.6]],[[15.9,.3],[15.9 ,2.6]],\n [[17.1,.3],[17.1,2.6]],[[19.9,.3],[19.9,2.6]]\n ],colo r=black):\nt1 := plots[textplot]([[0,3,`a`],[1,3,`b`],[2,3,`c`],[3,3,` d`],\n [0,2,`e`],[1,2,`f`],[2,2,`g`],[3,2,`h`],\n [0,1,`k`],[1,1,` m`],[2,1,`n`],[3,1,`p`],\n [0,0,`q`],[1,0,`r`],[2,0,`s`],[3,0,`t`], \n [5.5,2.5,`f`],[6.5,2.5,`g`],[7.5,2.5,`h`],\n [5.5,1.5,`m`],[6 .5,1.5,`n`],[7.5,1.5,`p`],\n [5.5,.5,`r`],[6.5,.5,`s`],[7.5,.5,`t`] ,\n [9.5,2.5,`e`],[10.5,2.5,`g`],[11.5,2.5,`h`],\n [9.5,1.5,`k`] ,[10.5,1.5,`n`],[11.5,1.5,`p`],\n [9.5,.5,`q`],[10.5,.5,`s`],[11.5, .5,`t`],\n [13.5,2.5,`e`],[14.5,2.5,`f`],[15.5,2.5,`h`],\n [13.5 ,1.5,`k`],[14.5,1.5,`m`],[15.5,1.5,`p`],\n [13.5,.5,`q`],[14.5,.5,` r`],[15.5,.5,`t`],\n [17.5,2.5,`e`],[18.5,2.5,`f`],[19.5,2.5,`g`], \n [17.5,1.5,`k`],[18.5,1.5,`m`],[19.5,1.5,`n`],\n [17.5,.5,`q`] ,[18.5,.5,`r`],[19.5,.5,`s`],\n [4.1,1.46,`=`],[4.7,1.5,`a`],[8.6,1 .5,`- b`],\n [12.6,1.5,`+ c`],[16.6,1.5,`- d`]],font=[TIMES,ITALIC,12 ],color=black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "p1 := plo t([[[-.4,-.2],[-.4,3.1]],[[3.4,-.2],[3.4,3.1]],\n [[5.6,.3],[5.6,2. 6]],[[8.4,.3],[8.4,2.6]]],color=black):\nt1 := plots[textplot]([[0,3,` a`],[1,3,`b`],[2,3,`c`],[3,3,`d`],\n [0,2,`e`],[1,2,`f`],[2,2,`g`], [3,2,`h`],\n [0,1,`k`],[1,1,`m`],[2,1,`n`],[3,1,`p`],\n [0,0,`q` ],[1,0,`r`],[2,0,`s`],[3,0,`t`],\n [6,2.5,`f`],[7,2.5,`g`],[8,2.5,` h`],\n [6,1.5,`m`],[7,1.5,`n`],[8,1.5,`p`],\n [6,.5,`r`],[7,.5,` s`],[8,.5,`t`],\n [5.2,1.5,`+ a`]],font=[TIMES,ITALIC,12],color=bla ck):\np2 := plottools[arrow]([3.7,1.5],[4.7,1.5],0,.13,.17,arrow,color =red):\np3 := plots[polygonplot]([[[-.2,-.2],[-.2,3.1],[.2,3.1],[.2,-. 2]],\n [[-.2,3.1],[3.2,3.1],[3.2,2.8],[-.2,2.8]]],\n style=pat chnogrid,color=COLOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot( [seq([.2*cos(i*dd),2.95+.2*sin(i*dd)],i=0..20)],color=red): \nplots[di splay]([p1,p2,p3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 870 "p1 := plot([[[-.4,-.2],[- .4,3.1]],[[3.4,-.2],[3.4,3.1]],\n [[5.6,.3],[5.6,2.6]],[[8.4,.3],[8 .4,2.6]]],color=black):\nt1 := plots[textplot]([[0,3,`a`],[1,3,`b`],[2 ,3,`c`],[3,3,`d`],\n [0,2,`e`],[1,2,`f`],[2,2,`g`],[3,2,`h`],\n \+ [0,1,`k`],[1,1,`m`],[2,1,`n`],[3,1,`p`],\n [0,0,`q`],[1,0,`r`],[2,0 ,`s`],[3,0,`t`],\n [6,2.5,`e`],[7,2.5,`g`],[8,2.5,`h`],\n [6,1.5 ,`k`],[7,1.5,`n`],[8,1.5,`p`],\n [6,.5,`q`],[7,.5,`s`],[8,.5,`t`], \n [5.2,1.5,`- b`]],font=[TIMES,ITALIC,12],color=black):\np2 := plo ttools[arrow]([3.7,1.5],[4.7,1.5],0,.13,.17,arrow,color=red):\np3 := p lots[polygonplot]([[[.8,-.2],[.8,3.1],[1.2,3.1],[1.2,-.2]],\n [[-. 2,3.1],[3.2,3.1],[3.2,2.8],[-.2,2.8]]],\n style=patchnogrid,color= COLOR(RGB,.9,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([1+.2*cos( i*dd),2.95+.2*sin(i*dd)],i=0..20)],color=red): \nplots[display]([p1,p2 ,p3,p4,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 872 "p1 := plot([[[-.4,-.2],[-.4,3.1]], [[3.4,-.2],[3.4,3.1]],\n [[5.6,.3],[5.6,2.6]],[[8.4,.3],[8.4,2.6]]] ,color=black):\nt1 := plots[textplot]([[0,3,`a`],[1,3,`b`],[2,3,`c`],[ 3,3,`d`],\n [0,2,`e`],[1,2,`f`],[2,2,`g`],[3,2,`h`],\n [0,1,`k`] ,[1,1,`m`],[2,1,`n`],[3,1,`p`],\n [0,0,`q`],[1,0,`r`],[2,0,`s`],[3, 0,`t`],\n [6,2.5,`e`],[7,2.5,`f`],[8,2.5,`h`],\n [6,1.5,`k`],[7, 1.5,`m`],[8,1.5,`p`],\n [6,.5,`q`],[7,.5,`r`],[8,.5,`t`],\n [5.2 ,1.5,`+ c`]],font=[TIMES,ITALIC,12],color=black):\np2 := plottools[arr ow]([3.7,1.5],[4.7,1.5],0,.13,.17,arrow,color=red):\np3 := plots[polyg onplot]([[[1.8,-.2],[1.8,3.1],[2.2,3.1],[2.2,-.2]],\n [[-.2,3.1],[ 3.2,3.1],[3.2,2.8],[-.2,2.8]]],\n style=patchnogrid,color=COLOR(RG B,.9,.9,.9)):\ndd := evalf(Pi/10):\np4 := plot([seq([2+.2*cos(i*dd),2. 95+.2*sin(i*dd)],i=0..20)],color=red): \nplots[display]([p1,p2,p3,p4,t 1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 872 "p1 := plot([[[-.4,-.2],[-.4,3.1]],[[3.4,-.2], [3.4,3.1]],\n [[5.6,.3],[5.6,2.6]],[[8.4,.3],[8.4,2.6]]],color=blac k):\nt1 := plots[textplot]([[0,3,`a`],[1,3,`b`],[2,3,`c`],[3,3,`d`],\n [0,2,`e`],[1,2,`f`],[2,2,`g`],[3,2,`h`],\n [0,1,`k`],[1,1,`m`], [2,1,`n`],[3,1,`p`],\n [0,0,`q`],[1,0,`r`],[2,0,`s`],[3,0,`t`],\n \+ [6,2.5,`e`],[7,2.5,`f`],[8,2.5,`g`],\n [6,1.5,`k`],[7,1.5,`m`],[8 ,1.5,`n`],\n [6,.5,`q`],[7,.5,`r`],[8,.5,`s`],\n [5.2,1.5,`- d`] ],font=[TIMES,ITALIC,12],color=black):\np2 := plottools[arrow]([3.7,1. 5],[4.7,1.5],0,.13,.17,arrow,color=red):\np3 := plots[polygonplot]([[[ 2.8,-.2],[2.8,3.1],[3.2,3.1],[3.2,-.2]],\n [[-.2,3.1],[3.2,3.1],[3 .2,2.8],[-.2,2.8]]],\n style=patchnogrid,color=COLOR(RGB,.9,.9,.9) ):\ndd := evalf(Pi/10):\np4 := plot([seq([3+.2*cos(i*dd),2.95+.2*sin(i *dd)],i=0..20)],color=red): \nplots[display]([p1,p2,p3,p4,t1],axes=non e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "c hequerboard of signs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "p1 \+ := plot([[[-.4,-.2],[-.4,3.1]],[[3.4,-.2],[3.4,3.1]]],color=black):\nt 1 := plots[textplot]([[0,3,`+`],[1,3,`-`],[2,3,`+`],[3,3,`-`],\n [0, 2,`-`],[1,2,`+`],[2,2,`-`],[3,2,`+`],\n [0,1,`+`],[1,1,`-`],[2,1,`+` ],[3,1,`-`],\n [0,0,`-`],[1,0,`+`],[2,0,`-`],[3,0,`+`]],font=[TIMES,B OLD,15],color=black):\np2 := plottools[arrow]([2.7,1],[3.7,1],0,.13,.1 7,arrow,color=red): \nplots[display]([p1,t1],axes=none);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 " 4 x 4 determinant example " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " matrix([[0,1,4,5],[3,1,2,1],[0,1,3,0],[2,1,0,3]] ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "AA := Matrix([[0,1,4,5],[3,1,2,1],[0,1,3,0],[2,1,0,3] ]);\nLinearAlgebra[Determinant](AA);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#AAG-%'RTABLEG6%\"*W09X\"-%'MATRIXG6#7&7&\"\"!\"\"\"\"\"%\"\"&7&\" \"$F/\"\"#F/7&F.F/F3F.7&F4F/F.F3%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#U" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 772 "p1 := plot([[[-.4,-.2],[-.4,3.1]],[[3.4, -.2],[3.4,3.1]],\n [[5.1,.3],[5.1,2.6]],[[7.9,.3],[7.9,2.6]],\n \+ [[9.1,.3],[9.1,2.6]],[[11.9,.3],[11.9,2.6]]\n ],color=black):\nt1 : = plots[textplot]([[0,3,`0`],[1,3,`1`],[2,3,`4`],[3,3,`5`],\n [0,2,` 3`],[1,2,`1`],[2,2,`2`],[3,2,`1`],\n [0,1,`0`],[1,1,`1`],[2,1,`3`],[ 3,1,`0`],\n [0,0,`2`],[1,0,`1`],[2,0,`0`],[3,0,`3`],\n [5.5,2.5,` 1`],[6.5,2.5,`4`],[7.5,2.5,`5`],\n [5.5,1.5,`1`],[6.5,1.5,`3`],[7.5 ,1.5,`0`],\n [5.5,.5,`1`],[6.5,.5,`0`],[7.5,.5,`3`],\n [9.5,2.5, `1`],[10.5,2.5,`4`],[11.5,2.5,`5`],\n [9.5,1.5,`1`],[10.5,1.5,`2`], [11.5,1.5,`1`],\n [9.5,.5,`1`],[10.5,.5,`3`],[11.5,.5,`0`],\n [4 .1,1.46,`=`],[4.7,1.5,`- 3`],[8.6,1.5,`- 2`]],font=[TIMES,ROMAN,12],co lor=black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1544 "A := matri x([[1,4,5], [1,2,1], [1,3,0]]);\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2 .4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]] ,[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]], \n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],font=[TIMES,ROMAN,12],colo r=black):\nt2 := plots[textplot]([[3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1 ]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[3,2]]],font=[TIMES,ROMAN,12],co lor=COLOR(RGB,0,.6,0)):\nt3 := plots[textplot]([[3,3,A[3,1]*A[2,2]*A[1 ,3]],[4,3,A[3,2]*A[2,3]*A[1,1]],\n [5,3,A[3,3]* A[2,1]*A[1,2]]],\n f ont=[TIMES,ROMAN,12],color=COLOR(RGB,0,0,.9)):\nt4 := plots[textplot]( [[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1 ,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,.95,0,0) ):\np2 := plottools[arrow]([-.01,-.01],[2.7,2.7],0,.15,.05,arrow,\n \+ color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np3 := plottools[ arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6, .6,1),thickness=2,linestyle=2):\np4 := plottools[arrow]([1.99,-.01],[4 .7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,li nestyle=2):\np5 := plottools[arrow]([-.01,1.99],[2.7,-.702],0,.15,.05, arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702],0,.15,.05,arrow,\n color= COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np7 := plottools[arrow]([ 1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3) ,thickness=2,linestyle=2):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2 ,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 841 "p1 := plo t([[[-.4,-.2],[-.4,3.1]],[[3.4,-.2],[3.4,3.1]],\n [[4.6,-.2],[4.6,3 .1]],[[8.4,-.2],[8.4,3.1]],\n [[9.6,-.2],[9.6,3.1]],[[13.4,-.2],[13 .4,3.1]]\n ],color=black):\nt1 := plots[textplot]([[0,3,`0`],[1,3,` 1`],[2,3,`4`],[3,3,`5`],\n [0,2,`3`],[1,2,`1`],[2,2,`2`],[3,2,`1`], \n [0,1,`0`],[1,1,`1`],[2,1,`3`],[3,1,`0`],\n [0,0,`2`],[1,0,`1`], [2,0,`0`],[3,0,`3`],\n [5,3,`0`],[6,3,`1`],[7,3,`4`],[8,3,`5`],\n \+ [5,2,`1`],[6,2,`0`],[7,2,`2`],[8,2,`-2`],\n [5,1,`0`],[6,1,`1`],[7,1 ,`3`],[8,1,`0`],\n [5,0,`2`],[6,0,`1`],[7,0,`0`],[8,0,`3`],\n [10, 3,`0`],[11,3,`1`],[12,3,`4`],[13,3,`5`],\n [10,2,`1`],[11,2,`0`],[12 ,2,`2`],[13,2,`-2`],\n [10,1,`0`],[11,1,`1`],[12,1,`3`],[13,1,`0`], \n [10,0,`0`],[11,0,`1`],[12,0,`-4`],[13,0,`7`], [4,1.5,`=`],[9,1 .5,`=`]],font=[TIMES,ROMAN,12],color=black):\nplots[display]([p1,t1],a xes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 570 "p1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4, 2.1]],\n [[3.6,-.2],[3.6,2.1]],[[6.4,-.2],[6.4,2.1]],\n [[7.6,. 3],[7.6,1.6]],[[9.4,.3],[9.4,1.6]]],color=black):\nt1 := plots[textplo t]([[0,2,`1`],[1,2,`4`],[2,2,`5`],\n [0,1,`1`],[1,1,`3`],[2,1,`0`], \n [0,0,`1`],[1,0,`-4`],[2,0,`7`],\n [4,2,`1`],[5,2,`4`],[6,2,`5`] ,\n [4,1,`0`],[5,1,`-1`],[6,1,`-5`],\n [4,0,`0`],[5,0,`-8`],[6,0,` 2`],\n [8,1.5,`-1`],[9,1.5,`-5`],[8,.5,`-8`],[9,.5,`2`],\n [-.7,1, `-`],[3.1,1,`= -`],[7.1,1,`= -`]],\n font=[TIMES,ROMAN,12],color= black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 145112148 145116516 145119264 145123308 145125084 145126536 145130932 145133680 145135132 145140544 }{RTABLE M7R0 I6RTABLE_SAVE/145112148X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""#""$F(F'""%!""""&F' 6" } {RTABLE M7R0 I6RTABLE_SAVE/145116516X,%)anythingG6"6"[gl!"%!!!#*"$"$#!#>""##"#8F)#""&F)!"$F) """#"#6F)#!"(F)#F.F)6" } {RTABLE M7R0 I6RTABLE_SAVE/145119264X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""%""!F'""$!""""&""'F '6" } {RTABLE M7R0 I6RTABLE_SAVE/145123308X,%)anythingG6"6"[gl!"%!!!#*"$"$""#""%"")F(F'""(""&""!!" #6" } {RTABLE M7R0 I6RTABLE_SAVE/145125084X,%)anythingG6"6"[gl!"%!!!#*"$"$!""""%""$""#""&!"$F)!"#F *6" } {RTABLE M7R0 I6RTABLE_SAVE/145126536X,%)anythingG6"6"[gl!"%!!!#1"%"%"""F'""#F'F(""$F)""&F)F* ""'"")""%""(F."#?6" } {RTABLE M7R0 I6RTABLE_SAVE/145130932X,%)anythingG6"6"[gl!"%!!!#*"$"$""#""$""!!"""""F'F+F*F'6 " } {RTABLE M7R0 I6RTABLE_SAVE/145133680X,%)anythingG6"6"[gl!"%!!!#*"$"$""#""%""!"""""$!""""&"") F)6" } {RTABLE M7R0 I6RTABLE_SAVE/145135132X,%)anythingG6"6"[gl!"%!!!#*"$"$"""F'""#""!F'F'""&F(!""6 " } {RTABLE M7R0 I6RTABLE_SAVE/145140544X,%)anythingG6"6"[gl!"%!!!#1"%"%""!""$F'""#"""F*F*F*""%F )F(F'""&F*F'F(6" }