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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Gaussian elimination and Gauss-Jo
rdan elimination " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana
imo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  30.3.2007
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 90 "Solving a system of linear equati
ons by Gaussian elimination or Gauss-Jordan elimination  " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "C
onsider the problem of solving the system of equations: " }}{PARA 256 
"" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([2*x+y-z=2,``],[4*x-
y-3*z=-2,``],[2*x+2*y+z=9,``])" "6#-%*PIECEWISEG6%7$/,(*&\"\"#\"\"\"%
\"xGF+F+%\"yGF+%\"zG!\"\"F*%!G7$/,(*&\"\"%F+F,F+F+F-F/*&\"\"$F+F.F+F/,
$F*F/F07$/,(*&F*F+F,F+F+*&F*F+F-F+F+F.F+\"\"*F0" }{TEXT -1 13 "-------
 (i). " }}{PARA 0 "" 0 "" {TEXT -1 51 "This system can be represented \+
in the matrix form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "matrix([[2,1,-1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z]])=matrix([
[2],[-2],[9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\"7%\"\"%,$F+F
-,$\"\"$F-7%F*F*F+F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7#F*7#,$F*F-7
#\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 154 "The standard method of solution by manipulating the
 original system of equations can be simplified by working with the co
efficients only as entries in an " }{TEXT 259 16 "augmented matrix" }
{TEXT -1 14 " of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[2, 1, -1,`|`,2], [4, -1, -3,`|`,-2], [2, 2, 1,
`|`,9]])" "6#-%'matrixG6#7%7'\"\"#\"\"\",$F)!\"\"%\"|grGF(7'\"\"%,$F)F
+,$\"\"$F+F,,$F(F+7'F(F(F)F,\"\"*" }{TEXT -1 16 "   ------- (ii)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "Thus we \+
augment the cefficient matrix with an extra column formed by the const
ants from the right hand sides of the original equations." }}{PARA 0 "
" 0 "" {TEXT -1 386 "The process of eliminating a variable can be achi
eved with the original system of equations (i) by the standard operati
ons of multiplying the equations by suitable constants, and then addin
g or subtracting the equations. For example, subtracting the first equ
ation from the 3rd and subtracting 2 times the 1st equation from the 2
nd equation produces the equivalent system of equations: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*x+y-z = 2, ``]
,[-3*y-z = -6, ``],[y+2*z = 7, ``]);" "6#-%*PIECEWISEG6%7$/,(*&\"\"#\"
\"\"%\"xGF+F+%\"yGF+%\"zG!\"\"F*%!G7$/,&*&\"\"$F+F-F+F/F.F/,$\"\"'F/F0
7$/,&F-F+*&F*F+F.F+F+\"\"(F0" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 22 "in which the variable " }{TEXT 263 1 "x" }{TEXT -1 53 " h
as been eliminated from the 2nd and 3rd equations. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "These operations can be
 mirrored as suitable row operations applied to the augmented matrix (
ii)." }}{PARA 0 "" 0 "" {TEXT -1 69 "Thus we subtract row 1 from the r
ow 3, and 2 times row 1 from row 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([R[2]=R[
2]-2*R[1],``],[R[3]=R[3]-R[1],``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#
,&&F)6#F+\"\"\"*&F+F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F9F/&F)6#F/F
3F4" }{TEXT -1 4 "... " }{XPPEDIT 18 0 "matrix([[2, 1, -1, `|`, 2], [0
, -3, -1, `|`, -6], [0, 1, 2, `|`, 7]])" "6#-%'matrixG6#7%7'\"\"#\"\"
\",$F)!\"\"%\"|grGF(7'\"\"!,$\"\"$F+,$F)F+F,,$\"\"'F+7'F.F)F(F,\"\"(" 
}{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 98 "Now interchange rows 2 and 3, which corresponds to interc
hanging the corresponding two equations. " }}{PARA 256 "" 0 "" {TEXT 
-1 6 " swap " }{XPPEDIT 18 0 "R[2];" "6#&%\"RG6#\"\"#" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "R[3];" "6#&%\"RG6#\"\"$" }{TEXT -1 6 "  ... " }
{XPPEDIT 18 0 "matrix([[2, 1, -1, `|`, 2], [0, 1, 2, `|`, 7], [0, -3, \+
-1, `|`, -6]])" "6#-%'matrixG6#7%7'\"\"#\"\"\",$F)!\"\"%\"|grGF(7'\"\"
!F)F(F,\"\"(7'F.,$\"\"$F+,$F)F+F,,$\"\"'F+" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Multiply row 3 by
 -1. " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "R[3]=R[3]*`
.`*``(-1)" "6#/&%\"RG6#\"\"$*(&F%6#F'\"\"\"%\".GF+-%!G6#,$F+!\"\"F+" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[2, 1, -1, `|`, 2], [0, 1,
 2, `|`, 7], [0, 3, 1, `|`, 6]])" "6#-%'matrixG6#7%7'\"\"#\"\"\",$F)!
\"\"%\"|grGF(7'\"\"!F)F(F,\"\"(7'F.\"\"$F)F,\"\"'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Subtract \+
3 times row 2 from row 3 . . .  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "R[3]=R[3]-3*R[2]" "6#/&%\"RG6#\"\"$,&&F%6#F'\"\"\"*&F'F
+&F%6#\"\"#F+!\"\"" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[2, 1
, -1, `|`, 2], [0, 1 ,2, `|`, 7], [0, 0, -5, `|`, -15]])" "6#-%'matrix
G6#7%7'\"\"#\"\"\",$F)!\"\"%\"|grGF(7'\"\"!F)F(F,\"\"(7'F.F.,$\"\"&F+F
,,$\"#:F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 29 " . . . and multiply row 3 by " }{XPPEDIT 18 0 "-1/
5" "6#,$*&\"\"\"F%\"\"&!\"\"F'" }{TEXT -1 15 " (or divide by " }
{XPPEDIT 18 0 "-5" "6#,$\"\"&!\"\"" }{TEXT -1 8 ") . . . " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]*`.`*``(-1/5)" "6#
/&%\"RG6#\"\"$*(&F%6#F'\"\"\"%\".GF+-%!G6#,$*&F+F+\"\"&!\"\"F3F+" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[2, 1, -1, `|`, 2], [0, 1,
 2, `|`, 7], [0, 0, 1, `|`, 3]])" "6#-%'matrixG6#7%7'\"\"#\"\"\",$F)!
\"\"%\"|grGF(7'\"\"!F)F(F,\"\"(7'F.F.F)F,\"\"$" }{TEXT -1 15 " -------
 (iii) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
56 "The corresponding system of equations at this stage is: " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*x+y-z = 2, ``]
,[y+2*z = 7, ``],[z = 3, ``]);" "6#-%*PIECEWISEG6%7$/,(*&\"\"#\"\"\"%
\"xGF+F+%\"yGF+%\"zG!\"\"F*%!G7$/,&F-F+*&F*F+F.F+F+\"\"(F07$/F.\"\"$F0
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 54 "The solution of the system can now be found easily by " }
{TEXT 259 17 "back-substitution" }{TEXT -1 14 ", as follows: " }}
{PARA 0 "" 0 "" {TEXT -1 11 "Substitute " }{XPPEDIT 18 0 "z=3" "6#/%\"
zG\"\"$" }{TEXT -1 53 " from the 3rd equation into the 2nd equation to
 give " }{XPPEDIT 18 0 "y+6=7" "6#/,&%\"yG\"\"\"\"\"'F&\"\"(" }{TEXT 
-1 28 " from which it follows that " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"
\"\"" }{TEXT -1 22 ". Then substitute for " }{TEXT 264 1 "y" }{TEXT 
-1 5 " and " }{TEXT 265 1 "z" }{TEXT -1 29 " in the 1st equation to gi
ve " }{XPPEDIT 18 0 "2*x+1-3=2" "6#/,(*&\"\"#\"\"\"%\"xGF'F'F'F'\"\"$!
\"\"F&" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#
" }{TEXT -1 156 ". The application of  row operations leading to the a
ugmented matrix (iii), in which the left-hand square matrix part has 0
's below the diagonal, is called " }{TEXT 259 20 "Gaussian elimination
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 165 "An alternative way \+
to complete the solution is to continue to apply suitable row operatio
ns to the augmented matrix (iii) until the left-hand part is reduced t
o the " }{TEXT 259 11 "unit matrix" }{TEXT -1 17 ". This is called " }
{TEXT 259 24 "Gauss-Jordan elimination" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Add row 3 to row 1 a
nd subtract 2 times row 3 from row 2. " }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "PIECEWISE([R[1]=R[1]+R[3],``],[R[2]=R[2]-2*R[3],``])" "6#-%*PIEC
EWISEG6$7$/&%\"RG6#\"\"\",&&F)6#F+F+&F)6#\"\"$F+%!G7$/&F)6#\"\"#,&&F)6
#F7F+*&F7F+&F)6#F1F+!\"\"F2" }{TEXT -1 4 "... " }{XPPEDIT 18 0 "matrix
([[2, 1, 0, `|`, 5], [0, 1, 0, `|`, 1], [0, 0, 1, `|`, 3]])" "6#-%'mat
rixG6#7%7'\"\"#\"\"\"\"\"!%\"|grG\"\"&7'F*F)F*F+F)7'F*F*F)F+\"\"$" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "Now subtract row 2 from \+
row 1 . . . " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "R[1]
=R[1]-R[3]" "6#/&%\"RG6#\"\"\",&&F%6#F'F'&F%6#\"\"$!\"\"" }{TEXT -1 6 
"  ... " }{XPPEDIT 18 0 "matrix([[2, 0, 0, `|`, 4], [0, 1, 0, `|`, 1],
 [0, 0, 1, `|`, 3]])" "6#-%'matrixG6#7%7'\"\"#\"\"!F)%\"|grG\"\"%7'F)
\"\"\"F)F*F-7'F)F)F-F*\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 23 " and multiply row 1 by " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"
#!\"\"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "R[1]=R[1]*`.`*``(1/2)" "6#/&%\"RG6#\"\"\"*(&F%6#F'F'%\"
.GF'-%!G6#*&F'F'\"\"#!\"\"F'" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "mat
rix([[1, 0, 0, `|`, 2], [0, 1, 0, `|`, 1], [0, 0, 1, `|`, 3]]);" "6#-%
'matrixG6#7%7'\"\"\"\"\"!F)%\"|grG\"\"#7'F)F(F)F*F(7'F)F)F(F*\"\"$" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 58 "The corresponding system of equations is now the solution
 " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 3 ",  " }{XPPEDIT 
18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "z=3" "6#
/%\"zG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 146 "The solu
tion can be checked by substitution in the original system, or (which \+
amounts to the same thing) by performing the matrix multiplication: " 
}}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "matrix([[2, 1, -1]
, [4, -1, -3], [2, 2, 1]])*matrix([[2], [1], [3]]) = matrix([[2], [-2]
, [9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\"7%\"\"%,$F+F-,$\"\"
$F-7%F*F*F+F+-F&6#7%7#F*7#F+7#F2F+-F&6#7%7#F*7#,$F*F-7#\"\"*" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 14 ": The all
owed " }{TEXT 259 14 "row operations" }{TEXT -1 17 " are as follows: \+
" }}{PARA 15 "" 0 "" {TEXT -1 47 "A row can be multiplied by a non-zer
o constant." }}{PARA 15 "" 0 "" {TEXT -1 35 "Any two rows can be inter
changed.  " }}{PARA 15 "" 0 "" {TEXT -1 81 "A non-zero multiple of one
 row may be added to (or subtracted from) another row. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl
e 1 " }}{PARA 0 "" 0 "" {TEXT 276 8 "Question" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 38 "Solve the system of linear equations: " }
{XPPEDIT 18 0 "PIECEWISE([x-y-2*z = -2, ``],[3*x-y+z = 6, ``],[x-3*y-4
*z = -4, ``]);" "6#-%*PIECEWISEG6%7$/,(%\"xG\"\"\"%\"yG!\"\"*&\"\"#F*%
\"zGF*F,,$F.F,%!G7$/,(*&\"\"$F*F)F*F*F+F,F/F*\"\"'F17$/,(F)F**&F6F*F+F
*F,*&\"\"%F*F/F*F,,$F=F,F1" }{TEXT -1 62 "(a) by Gaussian elimination \+
 (b) by Gauss-Jordan elimination. " }}{PARA 0 "" 0 "" {TEXT 277 8 "Sol
ution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 77 "(a) Applying ro
w operations to the augmented matrix of the system we obtain: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, -1, -2, `
|`, -2], [3, -1, 1, `|`, 6], [1, -3, -4, `|`, -4]]);" "6#-%'matrixG6#7
%7'\"\"\",$F(!\"\",$\"\"#F*%\"|grG,$F,F*7'\"\"$,$F(F*F(F-\"\"'7'F(,$F0
F*,$\"\"%F*F-,$F6F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[
2]-3*R[1], ``],[R[3] = R[3]-R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#
\"\"#,&&F)6#F+\"\"\"*&\"\"$F/&F)6#F/F/!\"\"%!G7$/&F)6#F1,&&F)6#F1F/&F)
6#F/F4F5" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, -1, -2, `|`
, -2], [0, 2, 7, `|`, 12], [0, -2, -2, `|`, -2]]);" "6#-%'matrixG6#7%7
'\"\"\",$F(!\"\",$\"\"#F*%\"|grG,$F,F*7'\"\"!F,\"\"(F-\"#77'F0,$F,F*,$
F,F*F-,$F,F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]+R[2]" "6#/&%\"RG6
#\"\"$,&&F%6#F'\"\"\"&F%6#\"\"#F+" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 
"matrix([[1, -1, -2, `|`, -2], [0, 2, 7, `|`, 12], [0, 0, 5, `|`, 10]]
);" "6#-%'matrixG6#7%7'\"\"\",$F(!\"\",$\"\"#F*%\"|grG,$F,F*7'\"\"!F,
\"\"(F-\"#77'F0F0\"\"&F-\"#5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3] = R[3]
*``(1/5);" "6#/&%\"RG6#\"\"$*&&F%6#F'\"\"\"-%!G6#*&F+F+\"\"&!\"\"F+" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, -1, -2, `|`, -2], [0, \+
2, 7, `|`, 12], [0, 0, 1, `|`, 2]]);" "6#-%'matrixG6#7%7'\"\"\",$F(!\"
\",$\"\"#F*%\"|grG,$F,F*7'\"\"!F,\"\"(F-\"#77'F0F0F(F-F," }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The
 last augmented matrix corresponds to the system " }{XPPEDIT 18 0 "PIE
CEWISE([x-y-2*z = -2, ``],[2*y+7*z = 12, ``],[z = 2, ``]);" "6#-%*PIEC
EWISEG6%7$/,(%\"xG\"\"\"%\"yG!\"\"*&\"\"#F*%\"zGF*F,,$F.F,%!G7$/,&*&F.
F*F+F*F**&\"\"(F*F/F*F*\"#7F17$/F/F.F1" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 48 "This system can be solved by back substitution. " }}
{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "z = 2;" "6
#/%\"zG\"\"#" }{TEXT -1 27 " in the 2nd equation gives " }{XPPEDIT 18 
0 "2*y+14 = 12;" "6#/,&*&\"\"#\"\"\"%\"yGF'F'\"#9F'\"#7" }{TEXT -1 11 
", so that  " }{XPPEDIT 18 0 "y = -1;" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Then substituting for " }
{TEXT 278 1 "y" }{TEXT -1 5 " and " }{TEXT 279 1 "z" }{TEXT -1 27 " in
 the 1st equation gives " }{XPPEDIT 18 0 "x+1-4 = -2;" "6#/,(%\"xG\"\"
\"F&F&\"\"%!\"\",$\"\"#F(" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x
 = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
16 "The solution is " }{XPPEDIT 18 0 "x = 1,y = -1,z = 2;" "6%/%\"xG\"
\"\"/%\"yG,$F%!\"\"/%\"zG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" 
{TEXT -1 178 "Instead of using back-substitution, we can continue appl
ying row operations to the last augmented matrix in order to attempt t
o reduce the left-hand part to the identity matrix. " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, -1, -2, `|`, -2], [0, \+
2, 7, `|`, 12], [0, 0, 1, `|`, 2]]);" "6#-%'matrixG6#7%7'\"\"\",$F(!\"
\",$\"\"#F*%\"|grG,$F,F*7'\"\"!F,\"\"(F-\"#77'F0F0F(F-F," }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "R[2] = R[2]-7*R[3];" "6#/&%\"RG6#\"\"#,&&F%6#F'\"\"
\"*&\"\"(F+&F%6#\"\"$F+!\"\"" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "mat
rix([[1, -1, -2, `|`, -2], [0, 2, 0, `|`, -2], [0, 0, 1, `|`, 2]]);" "
6#-%'matrixG6#7%7'\"\"\",$F(!\"\",$\"\"#F*%\"|grG,$F,F*7'\"\"!F,F0F-,$
F,F*7'F0F0F(F-F," }{TEXT -1 2 "  " }}{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'\"\"\"-%!G6#*&F+F+F'!\"\"F+" }{TEXT -1 6 "  ..
. " }{XPPEDIT 18 0 "matrix([[1, -1, -2, `|`, -2], [0, 1, 0, `|`, -1], \+
[0, 0, 1, `|`, 2]])" "6#-%'matrixG6#7%7'\"\"\",$F(!\"\",$\"\"#F*%\"|gr
G,$F,F*7'\"\"!F(F0F-,$F(F*7'F0F0F(F-F," }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 
"R[1] = R[1]+R[2]+2*R[3]" "6#/&%\"RG6#\"\"\",(&F%6#F'F'&F%6#\"\"#F'*&F
-F'&F%6#\"\"$F'F'" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, 0,
0, `|`, 1], [0, 1, 0, `|`, -1], [0, 0, 1, `|`, 2]])" "6#-%'matrixG6#7%
7'\"\"\"\"\"!F)%\"|grGF(7'F)F(F)F*,$F(!\"\"7'F)F)F(F*\"\"#" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "T
his also gives the solution " }{XPPEDIT 18 0 "x = 1,y = -1,z = 2;" "6%
/%\"xG\"\"\"/%\"yG,$F%!\"\"/%\"zG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "eq1 := \+
x-y-2*z=-2;\neq2 := 3*x-y+z=6;\neq3 := x-3*y-4*z=-4;\nsolve(\{eq1,eq2,
eq3\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(%\"xG\"
\"\"%\"yG!\"\"*&\"\"#F(%\"zGF(F*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$eq2G/,(*&\"\"$\"\"\"%\"xGF)F)%\"yG!\"\"%\"zGF)\"\"'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$eq3G/,(%\"xG\"\"\"*&\"\"$F(%\"yGF(!\"\"*&\"\"
%F(%\"zGF(F,!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG\"\"\"/%\"
yG!\"\"/%\"zG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "
Example 2 " }}{PARA 0 "" 0 "" {TEXT 280 8 "Question" }{TEXT -1 2 ": " 
}}{PARA 0 "" 0 "" {TEXT -1 38 "Solve the system of linear equations: \+
" }{XPPEDIT 18 0 "PIECEWISE([2*x+6*y+z = 7, ``],[x+2*y-z = -1, ``],[5*
x+7*y-4*z = 9, ``]);" "6#-%*PIECEWISEG6%7$/,(*&\"\"#\"\"\"%\"xGF+F+*&
\"\"'F+%\"yGF+F+%\"zGF+\"\"(%!G7$/,(F,F+*&F*F+F/F+F+F0!\"\",$F+F7F27$/
,(*&\"\"&F+F,F+F+*&F1F+F/F+F+*&\"\"%F+F0F+F7\"\"*F2" }{TEXT -1 62 "(a)
 by Gaussian elimination  (b) by Gauss-Jordan elimination. " }}{PARA 
0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 77 "(a) Applying row operations to the augmented matrix of th
e system we obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "matrix([[2, 6, 1, `|`, 7], [1,2,-1, `|`, -1], [5,7,-4, `|`, 9]])" "
6#-%'matrixG6#7%7'\"\"#\"\"'\"\"\"%\"|grG\"\"(7'F*F(,$F*!\"\"F+,$F*F/7
'\"\"&F,,$\"\"%F/F+\"\"*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " swap " }{XPPEDIT 18 0 "R[1]" "6#
&%\"RG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R[2]" "6#&%\"RG6#
\"\"#" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,2,-1, `|`, -1],
[2, 6, 1, `|`, 7], [5,7,-4, `|`, 9]])" "6#-%'matrixG6#7%7'\"\"\"\"\"#,
$F(!\"\"%\"|grG,$F(F+7'F)\"\"'F(F,\"\"(7'\"\"&F0,$\"\"%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]-5*R[1],``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&F+
F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F9F/*&\"\"&F/&F)6#F/F/F3F4" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,2,-1, `|`, -1],[0,2,3, \+
`|`, 9], [0,-3,1, `|`, 14]])" "6#-%'matrixG6#7%7'\"\"\"\"\"#,$F(!\"\"%
\"|grG,$F(F+7'\"\"!F)\"\"$F,\"\"*7'F/,$F0F+F(F,\"#9" }{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'\"\"\"-%!G
6#*&F+F+F'!\"\"F+" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,2,-
1, `|`, -1],[0,1,3/2, `|`, 9/2], [0,-3,1, `|`, 14]])" "6#-%'matrixG6#7
%7'\"\"\"\"\"#,$F(!\"\"%\"|grG,$F(F+7'\"\"!F(*&\"\"$F(F)F+F,*&\"\"*F(F
)F+7'F/,$F1F+F(F,\"#9" }{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#\"\"$,&&F%6#F'\"\"\"*&F'F+&F%6#\"\"#F+F+" }{TEXT -1 6 
"  ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, -1], [0, 1, 3/2, `|`,
 9/2], [0, 0, 11/2, `|`, 55/2]]);" "6#-%'matrixG6#7%7'\"\"\"\"\"#,$F(!
\"\"%\"|grG,$F(F+7'\"\"!F(*&\"\"$F(F)F+F,*&\"\"*F(F)F+7'F/F/*&\"#6F(F)
F+F,*&\"#bF(F)F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]*``(2/11)" "
6#/&%\"RG6#\"\"$*&&F%6#F'\"\"\"-%!G6#*&\"\"#F+\"#6!\"\"F+" }{TEXT -1 
6 "  ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, -1], [0, 1, 3/2, `|
`, 9/2], [0, 0, 1, `|`, 5]]);" "6#-%'matrixG6#7%7'\"\"\"\"\"#,$F(!\"\"
%\"|grG,$F(F+7'\"\"!F(*&\"\"$F(F)F+F,*&\"\"*F(F)F+7'F/F/F(F,\"\"&" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 52 "The last augmented matrix corresponds to the system " }
{XPPEDIT 18 0 "PIECEWISE([x+2*y-z=-1,``],[y+3*z/2=9/2,``],[z=5,``])" "
6#-%*PIECEWISEG6%7$/,(%\"xG\"\"\"*&\"\"#F*%\"yGF*F*%\"zG!\"\",$F*F/%!G
7$/,&F-F**(\"\"$F*F.F*F,F/F**&\"\"*F*F,F/F17$/F.\"\"&F1" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 48 "This system can be solved by back s
ubstitution. " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }
{XPPEDIT 18 0 "z=5" "6#/%\"zG\"\"&" }{TEXT -1 27 " in the 2nd equation
 gives " }{XPPEDIT 18 0 "y+15/2=9/2" "6#/,&%\"yG\"\"\"*&\"#:F&\"\"#!\"
\"F&*&\"\"*F&F)F*" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "y=-3" "6#
/%\"yG,$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Th
en substituting for " }{TEXT 282 1 "y" }{TEXT -1 5 " and " }{TEXT 283 
1 "z" }{TEXT -1 27 " in the 1st equation gives " }{XPPEDIT 18 0 "x+2*`
`(-3)-5=-1" "6#/,(%\"xG\"\"\"*&\"\"#F&-%!G6#,$\"\"$!\"\"F&F&\"\"&F.,$F
&F." }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x=10" "6#/%\"xG\"#5" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The solution is " }
{XPPEDIT 18 0 "x=10, y=-3, z=5" "6%/%\"xG\"#5/%\"yG,$\"\"$!\"\"/%\"zG
\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 178 "Instead of usin
g back-substitution, we can continue applying row operations to the la
st augmented matrix in order to attempt to reduce the left-hand part t
o the identity matrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "matrix([[1,2,-1, `|`, -1],[0,1,3/2, `|`, 9/2], [0,0,1, `|`, 5]])
" "6#-%'matrixG6#7%7'\"\"\"\"\"#,$F(!\"\"%\"|grG,$F(F+7'\"\"!F(*&\"\"$
F(F)F+F,*&\"\"*F(F)F+7'F/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([[1,0,-4, `|`, -10],[0,1,3/
2, `|`, 9/2], [0,0,1, `|`, 5]])" "6#-%'matrixG6#7%7'\"\"\"\"\"!,$\"\"%
!\"\"%\"|grG,$\"#5F,7'F)F(*&\"\"$F(\"\"#F,F-*&\"\"*F(F3F,7'F)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]+4*R[3], ``],[
R[2] = R[2]-``(3/2)*R[3], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\",&
&F)6#F+F+*&\"\"%F+&F)6#\"\"$F+F+%!G7$/&F)6#\"\"#,&&F)6#F9F+*&-F46#*&F3
F+F9!\"\"F+&F)6#F3F+FAF4" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix(
[[1,0,0, `|`, 10],[0,1,0, `|`, -3], [0,0,1, `|`, 5]])" "6#-%'matrixG6#
7%7'\"\"\"\"\"!F)%\"|grG\"#57'F)F(F)F*,$\"\"$!\"\"7'F)F)F(F*\"\"&" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "This also gives the solu
tion " }{XPPEDIT 18 0 "x=10, y=-3, z=5" "6%/%\"xG\"#5/%\"yG,$\"\"$!\"
\"/%\"zG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "eq1 := 2*x+6*y+z=7;\neq2 := x+2*y-z=-1;\neq3 := 5*x+7*y-4*z=9;\nso
lve(\{eq1,eq2,eq3\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
eq1G/,(%\"xG\"\"#*&\"\"'\"\"\"%\"yGF+F+%\"zGF+\"\"(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$eq2G/,(%\"xG\"\"\"*&\"\"#F(%\"yGF(F(%\"zG!\"\"F-" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(%\"xG\"\"&*&\"\"(\"\"\"%\"
yGF+F+*&\"\"%F+%\"zGF+!\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%
/%\"yG!\"$/%\"zG\"\"&/%\"xG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 284 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 43 "Show that the system of
 linear equations:  " }{XPPEDIT 18 0 "PIECEWISE([x-y+z = 2, ``],[2*x+y
-z = 1, ``],[2*x-2*y+2*z = 5, ``]);" "6#-%*PIECEWISEG6%7$/,(%\"xG\"\"
\"%\"yG!\"\"%\"zGF*\"\"#%!G7$/,(*&F.F*F)F*F*F+F*F-F,F*F/7$/,(*&F.F*F)F
*F**&F.F*F+F*F,*&F.F*F-F*F*\"\"&F/" }{TEXT -1 18 "has no solution.  " 
}}{PARA 0 "" 0 "" {TEXT 285 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "
" 0 "" {TEXT -1 73 "Applying row operations to the augmented matrix of
 the system we obtain: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, -1, 1, `|`, 2], [2, \+
1, -1, `|`, 1], [2, -2, 2, `|`, 5]]);" "6#-%'matrixG6#7%7'\"\"\",$F(!
\"\"F(%\"|grG\"\"#7'F,F(,$F(F*F+F(7'F,,$F,F*F,F+\"\"&" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 "    \+
  " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-2*R[1], ``],[R[3] = R[3]-2*
R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&F+F/&F
)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F9F/*&F+F/&F)6#F/F/F3F4" }{TEXT -1 
6 "  ... " }{XPPEDIT 18 0 "matrix([[1, -1, 1, `|`, 2], [0, 3, -3, `|`,
 -3], [0, 0, 0, `|`, 1]]);" "6#-%'matrixG6#7%7'\"\"\",$F(!\"\"F(%\"|gr
G\"\"#7'\"\"!\"\"$,$F/F*F+,$F/F*7'F.F.F.F+F(" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The last \+
augmented matrix corresponds to the system:  " }{XPPEDIT 18 0 "PIECEWI
SE([x-y+z = 2, ``],[3*y-3*z = -3, ``],[0 = 1, ``]);" "6#-%*PIECEWISEG6
%7$/,(%\"xG\"\"\"%\"yG!\"\"%\"zGF*\"\"#%!G7$/,&*&\"\"$F*F+F*F**&F4F*F-
F*F,,$F4F,F/7$/\"\"!F*F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Since the last \"equation\" is an \+
" }{TEXT 259 16 "untrue statement" }{TEXT -1 62 ", this means that it \+
is impossible to find three real numbers " }{TEXT 286 1 "x" }{TEXT -1 
2 ", " }{TEXT 287 1 "y" }{TEXT -1 5 " and " }{TEXT 288 1 "z" }{TEXT 
-1 39 " to satisfy the given three equations. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Alternatively, note that \+
the third equation " }{XPPEDIT 18 0 "2*x-2*y+2*z = 5" "6#/,(*&\"\"#\"
\"\"%\"xGF'F'*&F&F'%\"yGF'!\"\"*&F&F'%\"zGF'F'\"\"&" }{TEXT -1 27 " ca
n be written in the form" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x-y+z=5/2" "6#/,(%\"xG\"\"\"%\"yG!\"\"%\"zGF&*&\"\"&F&
\"\"#F(" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 46 "
Compare this equation with the first equation " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "x-y+z = 2" "6#/,(%\"xG\"\"\"%\"yG!\"\"
%\"zGF&\"\"#" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Since the two equations (i) and
 (ii) have the same expression " }{XPPEDIT 18 0 "x-y+z" "6#,(%\"xG\"\"
\"%\"yG!\"\"%\"zGF%" }{TEXT -1 113 " on the left, but different number
s on the right side, we see that it is impossible to obtain real  real
 numbers " }{TEXT 289 1 "x" }{TEXT -1 2 ", " }{TEXT 290 1 "y" }{TEXT 
-1 5 " and " }{TEXT 291 1 "z" }{TEXT -1 86 "  which satisfy these two \+
equations simultaneously. The original system is said to be " }{TEXT 
259 12 "inconsistent" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "eq1 := x-y+z=2;\neq2 := \+
2*x+y-z=1;\neq3 := 2*x-2*y+2*z=5;\nsolve(\{eq1,eq2,eq3\},\{x,y,z\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(%\"xG\"\"\"%\"yG!\"\"%\"zG
F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,(*&\"\"#\"\"\"%\"x
GF)F)%\"yGF)%\"zG!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(
*&\"\"#\"\"\"%\"xGF)F)*&F(F)%\"yGF)!\"\"*&F(F)%\"zGF)F)\"\"&" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "
" 0 "" {TEXT 274 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 41 "Show that the system of linear equations:" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*x+y+5*z = 4, ``],[3
*x-2*y+2*z = 2, ``],[5*x-8*y-4*z = -2, ``]);" "6#-%*PIECEWISEG6%7$/,(*
&\"\"#\"\"\"%\"xGF+F+%\"yGF+*&\"\"&F+%\"zGF+F+\"\"%%!G7$/,(*&\"\"$F+F,
F+F+*&F*F+F-F+!\"\"*&F*F+F0F+F+F*F27$/,(*&F/F+F,F+F+*&\"\")F+F-F+F9*&F
1F+F0F+F9,$F*F9F2" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 79 "has infinitely many solutions, and descri
be these solutions in a suitable way. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 275 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 73 "Applying row operations to the augmented matrix o
f the system we obtain: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2, 1, 5, `|`, 4], [3, \+
-2, 2, `|`, 2], [5, -8, -4, `|`, -2]]);" "6#-%'matrixG6#7%7'\"\"#\"\"
\"\"\"&%\"|grG\"\"%7'\"\"$,$F(!\"\"F(F+F(7'F*,$\"\")F0,$F,F0F+,$F(F0" 
}{TEXT -1 16 "                " }{XPPEDIT 18 0 "R[1] = R[1]-R[2];" "6#
/&%\"RG6#\"\"\",&&F%6#F'F'&F%6#\"\"#!\"\"" }{TEXT -1 6 "  ... " }
{XPPEDIT 18 0 "matrix([[-1, 3, 3, `|`, 2], [3, -2, 2, `|`, 2], [5, -8,
 -4, `|`, -2]]);" "6#-%'matrixG6#7%7',$\"\"\"!\"\"\"\"$F+%\"|grG\"\"#7
'F+,$F-F*F-F,F-7'\"\"&,$\"\")F*,$\"\"%F*F,,$F-F*" }{TEXT -1 11 "      \+
     " }{XPPEDIT 18 0 "R[1] = -R[1];" "6#/&%\"RG6#\"\"\",$&F%6#F'!\"\"
" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, -3, -3, `|`, -2], [
3, -2, 2, `|`, 2], [5, -8, -4, `|`, -2]]);" "6#-%'matrixG6#7%7'\"\"\",
$\"\"$!\"\",$F*F+%\"|grG,$\"\"#F+7'F*,$F/F+F/F-F/7'\"\"&,$\"\")F+,$\"
\"%F+F-,$F/F+" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R
[2]-3*R[1], ``],[R[3] = R[3]-5*R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"R
G6#\"\"#,&&F)6#F+\"\"\"*&\"\"$F/&F)6#F/F/!\"\"%!G7$/&F)6#F1,&&F)6#F1F/
*&\"\"&F/&F)6#F/F/F4F5" }{TEXT -1 7 "   ... " }{XPPEDIT 18 0 "matrix([
[1, -3, -3, `|`, -2], [0, 7, 11, `|`, 8], [0, 7, 11, `|`, 8]]);" "6#-%
'matrixG6#7%7'\"\"\",$\"\"$!\"\",$F*F+%\"|grG,$\"\"#F+7'\"\"!\"\"(\"#6
F-\"\")7'F1F2F3F-F4" }{TEXT -1 17 "                 " }{XPPEDIT 18 0 "
R[3] = R[3]-R[2];" "6#/&%\"RG6#\"\"$,&&F%6#F'\"\"\"&F%6#\"\"#!\"\"" }
{TEXT -1 5 " ... " }{XPPEDIT 18 0 "matrix([[1, -3, -3, `|`, -2], [0, 7
, 11, `|`, 8], [0, 0, 0, `|`, 0]]);" "6#-%'matrixG6#7%7'\"\"\",$\"\"$!
\"\",$F*F+%\"|grG,$\"\"#F+7'\"\"!\"\"(\"#6F-\"\")7'F1F1F1F-F1" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
71 "The last augmented matrix corresponds to the system of two equatio
ns:  " }{XPPEDIT 18 0 "PIECEWISE([x-3*y-3*z = -2, ``],[7*y+11*z = 8, `
`]);" "6#-%*PIECEWISEG6$7$/,(%\"xG\"\"\"*&\"\"$F*%\"yGF*!\"\"*&F,F*%\"
zGF*F.,$\"\"#F.%!G7$/,&*&\"\"(F*F-F*F**&\"#6F*F0F*F*\"\")F3" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
52 "We can find a solution as follows. Pick a value for " }{TEXT 295 
1 "z" }{TEXT -1 6 ", say " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }
{TEXT -1 9 ".  Then  " }{XPPEDIT 18 0 "7*y+11=8" "6#/,&*&\"\"(\"\"\"%
\"yGF'F'\"#6F'\"\")" }{TEXT -1 14 " which gives  " }{XPPEDIT 18 0 "7*y
=-3" "6#/*&\"\"(\"\"\"%\"yGF&,$\"\"$!\"\"" }{TEXT -1 11 ", so that  " 
}{XPPEDIT 18 0 "y=-3/7" "6#/%\"yG,$*&\"\"$\"\"\"\"\"(!\"\"F*" }{TEXT 
-1 22 ". Then, substituting  " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }
{TEXT -1 6 " and  " }{XPPEDIT 18 0 "y=-3/7" "6#/%\"yG,$*&\"\"$\"\"\"\"
\"(!\"\"F*" }{TEXT -1 19 "  in the equation  " }{XPPEDIT 18 0 "x-3*y-3
*z = -2" "6#/,(%\"xG\"\"\"*&\"\"$F&%\"yGF&!\"\"*&F(F&%\"zGF&F*,$\"\"#F
*" }{TEXT -1 8 " gives  " }{XPPEDIT 18 0 "x+9/7-3=-2" "6#/,(%\"xG\"\"
\"*&\"\"*F&\"\"(!\"\"F&\"\"$F*,$\"\"#F*" }{TEXT -1 11 ", so that  " }
{XPPEDIT 18 0 "x=1-9/7" "6#/%\"xG,&\"\"\"F&*&\"\"*F&\"\"(!\"\"F*" }
{XPPEDIT 18 0 "`` = -2/7;" "6#/%!G,$*&\"\"#\"\"\"\"\"(!\"\"F*" }{TEXT 
-1 30 ".  Thus we have the solution  " }{XPPEDIT 18 0 "x=-2/7, y=-3/7,
 z=1" "6%/%\"xG,$*&\"\"#\"\"\"\"\"(!\"\"F*/%\"yG,$*&\"\"$F(F)F*F*/%\"z
GF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 29 "However, we could have taken " }{TEXT 292 1 "z" }{TEXT 
-1 7 " to be " }{TEXT 259 15 "any real number" }{TEXT -1 1 " " }{TEXT 
293 1 "t" }{TEXT -1 8 ". Then  " }{XPPEDIT 18 0 "7*y+11*t=8" "6#/,&*&
\"\"(\"\"\"%\"yGF'F'*&\"#6F'%\"tGF'F'\"\")" }{TEXT -1 11 ", so that  \+
" }{XPPEDIT 18 0 "y=(8-11*t)/7" "6#/%\"yG*&,&\"\")\"\"\"*&\"#6F(%\"tGF
(!\"\"F(\"\"(F," }{TEXT -1 22 ". Then, substituting  " }{XPPEDIT 18 0 
"z = t;" "6#/%\"zG%\"tG" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "y=(8-11*
t)/7" "6#/%\"yG*&,&\"\")\"\"\"*&\"#6F(%\"tGF(!\"\"F(\"\"(F," }{TEXT 
-1 18 " in the equation  " }{XPPEDIT 18 0 "x-3*y-3*z = -2" "6#/,(%\"xG
\"\"\"*&\"\"$F&%\"yGF&!\"\"*&F(F&%\"zGF&F*,$\"\"#F*" }{TEXT -1 7 " giv
es " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-3*``((8-11*t
)/7)-3*t=-2" "6#/,(%\"xG\"\"\"*&\"\"$F&-%!G6#*&,&\"\")F&*&\"#6F&%\"tGF
&!\"\"F&\"\"(F2F&F2*&F(F&F1F&F2,$\"\"#F2" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x-24/7+33*t/7-3*t=-2" "6#/,*%\"xG\"\"\"*&\"#CF&\"\"(!\"
\"F**(\"#LF&%\"tGF&F)F*F&*&\"\"$F&F-F&F*,$\"\"#F*" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "x+12*t/7=10/7" "6#/,&%\"xG\"\"\"*(\"#7F&%\"tGF&\"\"(!\"
\"F&*&\"#5F&F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This
 gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=(10-12*t
)/7" "6#/%\"xG*&,&\"#5\"\"\"*&\"#7F(%\"tGF(!\"\"F(\"\"(F," }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Thus we have solutions: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = (10-12
*t)/7, ``],[y = (8-11*t)/7, ``],[z = t, ``]);" "6#-%*PIECEWISEG6%7$/%
\"xG*&,&\"#5\"\"\"*&\"#7F,%\"tGF,!\"\"F,\"\"(F0%!G7$/%\"yG*&,&\"\")F,*
&\"#6F,F/F,F0F,F1F0F27$/%\"zGF/F2" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 18 "for any choice of " }{TEXT 294 1 "t" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 54 "We conclude that the original system of e
quations has " }{TEXT 259 25 "infinitely many solutions" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Not
e" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "Multiplying the 2nd
 equation  " }{XPPEDIT 18 0 "3*x-2*y+2*z = 2" "6#/,(*&\"\"$\"\"\"%\"xG
F'F'*&\"\"#F'%\"yGF'!\"\"*&F*F'%\"zGF'F'F*" }{TEXT -1 12 " by 3 gives \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "9*x-6*y+6*z = 6" 
"6#/,(*&\"\"*\"\"\"%\"xGF'F'*&\"\"'F'%\"yGF'!\"\"*&F*F'%\"zGF'F'F*" }
{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 38 "while mult
iplying the first equation  " }{XPPEDIT 18 0 "2*x+y+5*z = 4" "6#/,(*&
\"\"#\"\"\"%\"xGF'F'%\"yGF'*&\"\"&F'%\"zGF'F'\"\"%" }{TEXT -1 13 "  by
 2 gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*x+2*y+
10*z = 8" "6#/,(*&\"\"%\"\"\"%\"xGF'F'*&\"\"#F'%\"yGF'F'*&\"#5F'%\"zGF
'F'\"\")" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 
36 "Now subtracting (ii) from (i) gives " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "5*x-8*y-4*z = -2" "6#/,(*&\"\"&\"\"\"%\"xGF'F
'*&\"\")F'%\"yGF'!\"\"*&\"\"%F'%\"zGF'F,,$\"\"#F," }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 27 "which is the 3rd equation. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "This shows that t
he third equation can be obtained from the first two equations and the
refore does not provide any new information beyond that already given \+
by the first two equations. " }}{PARA 0 "" 0 "" {TEXT -1 39 "We say th
at the system of equations is " }{TEXT 259 9 "dependent" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "eq1 := 2*x+y+5*z=4;\neq2 := \+
3*x-2*y+2*z=2;\neq3 := 5*x-8*y-4*z=-2;\nsolve(\{eq1,eq2,eq3\},\{x,y,z
\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(*&\"\"#\"\"\"%\"xGF)
F)%\"yGF)*&\"\"&F)%\"zGF)F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$eq2G/,(*&\"\"$\"\"\"%\"xGF)F)*&\"\"#F)%\"yGF)!\"\"*&F,F)%\"zGF)F)F," 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(*&\"\"&\"\"\"%\"xGF)F)*&\"
\")F)%\"yGF)!\"\"*&\"\"%F)%\"zGF)F.!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<%/%\"yG,&#\"\")\"\"(\"\"\"*(\"#6F*F)!\"\"%\"zGF*F-/%\"xG,&#\"#5
F)F**(\"#7F*F)F-F.F*F-/F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Calcula
tion of inverse matrices by Gauss-Jordan elimination " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 
"Introductory example .. the inverse of a 2 " }{TEXT 267 1 "x" }{TEXT 
-1 10 " 2 matrix " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 53 "Consider the problem of finding the inver
se of the 2 " }{TEXT 266 1 "x" }{TEXT -1 11 " 2 matrix  " }{XPPEDIT 
18 0 "A=matrix([[-3,6],[4,5]])" "6#/%\"AG-%'matrixG6#7$7$,$\"\"$!\"\"
\"\"'7$\"\"%\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Let
 the inverse matrix be " }{XPPEDIT 18 0 "matrix([[p,q],[r,s]])" "6#-%'
matrixG6#7$7$%\"pG%\"qG7$%\"rG%\"sG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "matrix([[-3, 6], [4, 5]])*matrix(
[[p,q],[r,s]])=matrix([[1,0], [0,1]])" "6#/*&-%'matrixG6#7$7$,$\"\"$!
\"\"\"\"'7$\"\"%\"\"&\"\"\"-F&6#7$7$%\"pG%\"qG7$%\"rG%\"sGF1-F&6#7$7$F
1\"\"!7$F?F1" }{TEXT -1 11 ", so that  " }{XPPEDIT 18 0 "matrix([[-3, \+
6], [4, 5]])*matrix([[p],[r]])=matrix([[1],[0]])" "6#/*&-%'matrixG6#7$
7$,$\"\"$!\"\"\"\"'7$\"\"%\"\"&\"\"\"-F&6#7$7#%\"pG7#%\"rGF1-F&6#7$7#F
17#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "matrix([[-3, 6], [4, 5]]
)*matrix([[q],[s]])=matrix([[0], [1]])" "6#/*&-%'matrixG6#7$7$,$\"\"$!
\"\"\"\"'7$\"\"%\"\"&\"\"\"-F&6#7$7#%\"qG7#%\"sGF1-F&6#7$7#\"\"!7#F1" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "Thus we need to solve \+
the two systems of equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([-3*p+6*r=1,``],[4*p+5*r=0,``])" "6#-%*PIECEW
ISEG6$7$/,&*&\"\"$\"\"\"%\"pGF+!\"\"*&\"\"'F+%\"rGF+F+F+%!G7$/,&*&\"\"
%F+F,F+F+*&\"\"&F+F0F+F+\"\"!F1" }{TEXT -1 11 "and        " }{XPPEDIT 
18 0 "PIECEWISE([-3*q+6*s=0,``],[4*q+5*s=1,``])" "6#-%*PIECEWISEG6$7$/
,&*&\"\"$\"\"\"%\"qGF+!\"\"*&\"\"'F+%\"sGF+F+\"\"!%!G7$/,&*&\"\"%F+F,F
+F+*&\"\"&F+F0F+F+F+F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 
"for the entries " }{XPPEDIT 18 0 "p,q,r,s" "6&%\"pG%\"qG%\"rG%\"sG" }
{TEXT -1 24 " of the inverse matrix. " }}{PARA 0 "" 0 "" {TEXT -1 109 
"These two systems can be solved simultaneously by applying Gauss-Jord
an elimination to the augnented matrix: " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "matrix([[-3, 6,`|`,1,0], [4, 5,`|`,0,1]])" "6
#-%'matrixG6#7$7',$\"\"$!\"\"\"\"'%\"|grG\"\"\"\"\"!7'\"\"%\"\"&F,F.F-
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "We may procede as fo
llows: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1]=R[1]*
``(1/3)" "6#/&%\"RG6#\"\"\"*&&F%6#F'F'-%!G6#*&F'F'\"\"$!\"\"F'" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[-1, 2,`|`,1/3,0], [4, 5,`
|`,0,1]])" "6#-%'matrixG6#7$7',$\"\"\"!\"\"\"\"#%\"|grG*&F)F)\"\"$F*\"
\"!7'\"\"%\"\"&F,F/F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]+4*R[1]
" "6#/&%\"RG6#\"\"$,&&F%6#F'\"\"\"*&\"\"%F+&F%6#F+F+F+" }{TEXT -1 6 " \+
 ... " }{XPPEDIT 18 0 "matrix([[-1, 2,`|`,1/3,0], [0, 13,`|`,4/3,1]])
" "6#-%'matrixG6#7$7',$\"\"\"!\"\"\"\"#%\"|grG*&F)F)\"\"$F*\"\"!7'F/\"
#8F,*&\"\"%F)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[3]*``(1/13),``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\",$&
F)6#F+!\"\"%!G7$/&F)6#\"\"$*&&F)6#F5F+-F06#*&F+F+\"#8F/F+F0" }{TEXT 
-1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, -2,`|`,-1/3,0], [0, 1,`|`,4/
39,1/13]])" "6#-%'matrixG6#7$7'\"\"\",$\"\"#!\"\"%\"|grG,$*&F(F(\"\"$F
+F+\"\"!7'F0F(F,*&\"\"%F(\"#RF+*&F(F(\"#8F+" }{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
'F'" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1, 0, `|`, -5/39, 2
/13], [0, 1, `|`, 4/39, 1/13]]);" "6#-%'matrixG6#7$7'\"\"\"\"\"!%\"|gr
G,$*&\"\"&F(\"#R!\"\"F/*&\"\"#F(\"#8F/7'F)F(F**&\"\"%F(F.F/*&F(F(F2F/
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 
0 "A^(-1) = matrix([[-5/39, 2/13], [4/39, 1/13]]);" "6#/)%\"AG,$\"\"\"
!\"\"-%'matrixG6#7$7$,$*&\"\"&F'\"#RF(F(*&\"\"#F'\"#8F(7$*&\"\"%F'F1F(
*&F'F'F4F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 52 "The result can be checked by matrix multiplicat
ion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*A^(-1)=`` \+
" "6#/*&%\"AG\"\"\")F%,$F&!\"\"F&%!G" }{XPPEDIT 18 0 "matrix([[-3, 6],
 [4, 5]])*matrix([[-5/39, 2/13], [4/39, 1/13]]) = matrix([[1,0], [0,1]
])" "6#/*&-%'matrixG6#7$7$,$\"\"$!\"\"\"\"'7$\"\"%\"\"&\"\"\"-F&6#7$7$
,$*&F0F1\"#RF,F,*&\"\"#F1\"#8F,7$*&F/F1F8F,*&F1F1F;F,F1-F&6#7$7$F1\"\"
!7$FCF1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 19 "or by using M
aple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "A := Matrix([[-3, 6], [4, 5]]);\nA^(-1);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")[C9?-%'MATRIXG6#7$7$!\"$\"
\"'7$\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")K'z.
#-%'MATRIXG6#7$7$#!\"&\"#R#\"\"#\"#87$#\"\"%F.#\"\"\"F1" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Example 2 .. the inverse of a 3 \+
" }{TEXT 268 1 "x" }{TEXT -1 10 " 3 matrix " }}{PARA 0 "" 0 "" {TEXT 
269 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 63 "Use G
auss-Jordan elimination to find the inverse of the matrix " }{XPPEDIT 
18 0 "A = matrix([[1,2,-1],[3,5,-1],[-2,-1,-2]])" "6#/%\"AG-%'matrixG6
#7%7%\"\"\"\"\"#,$F*!\"\"7%\"\"$\"\"&,$F*F-7%,$F+F-,$F*F-,$F+F-" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT -1 2 "
: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1,2,-1
,`|`,1,0,0],[3,5,-1,`|`,0,1,0],[-2,-1,-2,`|`,0,0,1]])" "6#-%'matrixG6#
7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F-7)\"\"$\"\"&,$F(F+F,F-F(F-7),$
F)F+,$F(F+,$F)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]-3*R[1],``],[R[3]=R[3]+2*R[1],``])" "6#-%*PIECEWISEG6$7$/&%\"RG6
#\"\"#,&&F)6#F+\"\"\"*&\"\"$F/&F)6#F/F/!\"\"%!G7$/&F)6#F1,&&F)6#F1F/*&
F+F/&F)6#F/F/F/F5" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,2,-
1,`|`,1,0,0],[0,-1,2,`|`,-3,1,0],[0,3,-4,`|`,2,0,1]])" "6#-%'matrixG6#
7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F-7)F-,$F(F+F)F,,$\"\"$F+F(F-7)F
-F1,$\"\"%F+F,F)F-F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2]=-R[2]" "6#/&%
\"RG6#\"\"#,$&F%6#F'!\"\"" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix
([[1,2,-1,`|`,1,0,0],[0,1,-2,`|`,3,-1,0],[0,3,-4,`|`,2,0,1]])" "6#-%'m
atrixG6#7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(\"\"!F-7)F-F(,$F)F+F,\"\"$,$F
(F+F-7)F-F0,$\"\"%F+F,F)F-F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]-3
*R[2]" "6#/&%\"RG6#\"\"$,&&F%6#F'\"\"\"*&F'F+&F%6#\"\"#F+!\"\"" }
{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,2,-1,`|`,1,0,0],[0,1,-2
,`|`,3,-1,0],[0,0,2,`|`,-7,3,1]])" "6#-%'matrixG6#7%7)\"\"\"\"\"#,$F(!
\"\"%\"|grGF(\"\"!F-7)F-F(,$F)F+F,\"\"$,$F(F+F-7)F-F-F)F,,$\"\"(F+F0F(
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "R[1]=R[1]-2*R[2]" "6#/&%\"RG6#\"\"\",&&
F%6#F'F'*&\"\"#F'&F%6#F,F'!\"\"" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "
matrix([[1,0,3,`|`,-5,2,0],[0,1,-2,`|`,3,-1,0],[0,0,2,`|`,-7,3,1]])" "
6#-%'matrixG6#7%7)\"\"\"\"\"!\"\"$%\"|grG,$\"\"&!\"\"\"\"#F)7)F)F(,$F/
F.F+F*,$F(F.F)7)F)F)F/F+,$\"\"(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]*``(1/2)" "6#/&%\"RG6#\"\"$*&&F%6#F'\"\"\"-%!G6#*&F+F+\"\"#!\"
\"F+" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,0,3,`|`,-5,2,0],
[0,1,-2,`|`,3,-1,0],[0,0,1,`|`,-7/2,3/2,1/2]])" "6#-%'matrixG6#7%7)\"
\"\"\"\"!\"\"$%\"|grG,$\"\"&!\"\"\"\"#F)7)F)F(,$F/F.F+F*,$F(F.F)7)F)F)
F(F+,$*&\"\"(F(F/F.F.*&F*F(F/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]+2*R[3]" "6#/&%\"RG6#\"\"#,&&F%6#F'\"\"\"*&F'F+&F%6#\"\"$F
+F+" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,0,3,`|`,-5,2,0],[
0,1,0,`|`,-4,2,1],[0,0,1,`|`,-7/2,3/2,1/2]])" "6#-%'matrixG6#7%7)\"\"
\"\"\"!\"\"$%\"|grG,$\"\"&!\"\"\"\"#F)7)F)F(F)F+,$\"\"%F.F/F(7)F)F)F(F
+,$*&\"\"(F(F/F.F.*&F*F(F/F.*&F(F(F/F." }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
R[1]=R[1]-3*R[3]" "6#/&%\"RG6#\"\"\",&&F%6#F'F'*&\"\"$F'&F%6#F,F'!\"\"
" }{TEXT -1 6 "  ... " }{XPPEDIT 18 0 "matrix([[1,0,0,`|`,11/2,-5/2,-3
/2],[0,1,0,`|`,-4,2,1],[0,0,1,`|`,-7/2,3/2,1/2]])" "6#-%'matrixG6#7%7)
\"\"\"\"\"!F)%\"|grG*&\"#6F(\"\"#!\"\",$*&\"\"&F(F-F.F.,$*&\"\"$F(F-F.
F.7)F)F(F)F*,$\"\"%F.F-F(7)F)F)F(F*,$*&\"\"(F(F-F.F.*&F4F(F-F.*&F(F(F-
F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Hence " }{XPPEDIT 18 0 "A^(-1) = matrix([[11/2, -5/2, -3/2
], [-4, 2, 1], [-7/2, 3/2, 1/2]]);" "6#/)%\"AG,$\"\"\"!\"\"-%'matrixG6
#7%7%*&\"#6F'\"\"#F(,$*&\"\"&F'F0F(F(,$*&\"\"$F'F0F(F(7%,$\"\"%F(F0F'7
%,$*&\"\"(F'F0F(F(*&F6F'F0F(*&F'F'F0F(" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The result can be ch
ecked by matrix multiplication:  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "A*A^(-1)=``" "6#/*&%\"AG\"\"\")F%,$F&!\"\"F&%!G" }
{XPPEDIT 18 0 "matrix([[1, 2, -1], [3, 5, -1], [-2, -1, -2]])*matrix([
[11/2, -5/2, -3/2], [-4, 2, 1], [-7/2, 3/2, 1/2]])=matrix([[1, 0, 0], \+
[ 0, 1, 0], [0, 0, 1]])" "6#/*&-%'matrixG6#7%7%\"\"\"\"\"#,$F*!\"\"7%
\"\"$\"\"&,$F*F-7%,$F+F-,$F*F-,$F+F-F*-F&6#7%7%*&\"#6F*F+F-,$*&F0F*F+F
-F-,$*&F/F*F+F-F-7%,$\"\"%F-F+F*7%,$*&\"\"(F*F+F-F-*&F/F*F+F-*&F*F*F+F
-F*-F&6#7%7%F*\"\"!FM7%FMF*FM7%FMFMF*" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 19 "or by using Maple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := Matrix([[1,2,-1],[3
,5,-1],[-2,-1,-2]]);\nA^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A
G-%'RTABLEG6$\")7[T?-%'MATRIXG6#7%7%\"\"\"\"\"#!\"\"7%\"\"$\"\"&F07%!
\"#F0F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")38m?-%'MATRI
XG6#7%7%#\"#6\"\"##!\"&F.#!\"$F.7%!\"%F.\"\"\"7%#!\"(F.#\"\"$F.#F5F." 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Example 3 .. the in
verse of a 3 " }{TEXT 273 1 "x" }{TEXT -1 10 " 3 matrix " }}{PARA 0 "
" 0 "" {TEXT 271 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 64 "Use Gauss-Jordan elimination to find the inverse of the m
atrix  " }{XPPEDIT 18 0 "A = matrix([[1, 2, -1], [3, 7, -5], [1, 2, 0]
]);" "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#,$F*!\"\"7%\"\"$\"\"(,$\"\"&F
-7%F*F+\"\"!" }{TEXT -1 4 " .  " }}{PARA 0 "" 0 "" {TEXT 272 8 "Soluti
on" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "matrix([[1, 2, -1, `|`, 1, 0, 0], [3, 7, -5, `|`, 0, 1, 0], [1, 2, \+
0, `|`, 0, 0, 1]]);" "6#-%'matrixG6#7%7)\"\"\"\"\"#,$F(!\"\"%\"|grGF(
\"\"!F-7)\"\"$\"\"(,$\"\"&F+F,F-F(F-7)F(F)F-F,F-F-F(" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-3*R[1], ``],[R[3] = R[3]-R[1], `
`])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&\"\"$F/&F)6#F/
F/!\"\"%!G7$/&F)6#F1,&&F)6#F1F/&F)6#F/F4F5" }{TEXT -1 6 " ...  " }
{XPPEDIT 18 0 "matrix([[1, 2, -1, `|`, 1, 0, 0], [0, 1, -2, `|`, -3, 1
, 0], [0, 0, 1, `|`, -1, 0, 1]]);" "6#-%'matrixG6#7%7)\"\"\"\"\"#,$F(!
\"\"%\"|grGF(\"\"!F-7)F-F(,$F)F+F,,$\"\"$F+F(F-7)F-F-F(F,,$F(F+F-F(" }
{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([R[1] = R[1]+R[3], ``],[R[2]
 = R[2]+2*R[3], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\",&&F)6#F+F+&
F)6#\"\"$F+%!G7$/&F)6#\"\"#,&&F)6#F7F+*&F7F+&F)6#F1F+F+F2" }{TEXT -1 
5 " ... " }{XPPEDIT 18 0 "matrix([[1,2,0,`|`,0,0,1],[0,1,0,`|`,-5,1,2]
,[0,0,1,`|`,-1,0,1]])" "6#-%'matrixG6#7%7)\"\"\"\"\"#\"\"!%\"|grGF*F*F
(7)F*F(F*F+,$\"\"&!\"\"F(F)7)F*F*F(F+,$F(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 14 "          ... " }{XPPEDIT 18 0 "matrix(
[[1,0,0,`|`,-10,-2,-3],[0,1,0,`|`,-5,1,2],[0,0,1,`|`,-1,0,1]])" "6#-%'
matrixG6#7%7)\"\"\"\"\"!F)%\"|grG,$\"#5!\"\",$\"\"#F-,$\"\"$F-7)F)F(F)
F*,$\"\"&F-F(F/7)F)F)F(F*,$F(F-F)F(" }{TEXT -1 3 "   " }}{PARA 0 "" 0 
"" {TEXT -1 7 "Hence  " }{XPPEDIT 18 0 "A^(-1)=matrix([[10, -2, -3], [
-5, 1, 2], [-1, 0, 1]])" "6#/)%\"AG,$\"\"\"!\"\"-%'matrixG6#7%7%\"#5,$
\"\"#F(,$\"\"$F(7%,$\"\"&F(F'F07%,$F'F(\"\"!F'" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 21 ": You can check that \+
" }{XPPEDIT 18 0 "A*`.`*A^(-1)=I" "6#/*(%\"AG\"\"\"%\".GF&)F%,$F&!\"\"
F&%\"IG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" 
{TEXT -1 102 "Solve the following system of equations (a) by Gaussian \+
elimination (b) by Gauss-Jordan elimination.  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x+2*y+3*z=13,``],[x+y-2*z=-2
,``],[x-3*y-4*z=-26,``])" "6#-%*PIECEWISEG6%7$/,(%\"xG\"\"\"*&\"\"#F*%
\"yGF*F**&\"\"$F*%\"zGF*F*\"#8%!G7$/,(F)F*F-F**&F,F*F0F*!\"\",$F,F7F27
$/,(F)F**&F/F*F-F*F7*&\"\"%F*F0F*F7,$\"#EF7F2" }{TEXT -1 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 -1 102 "Solve the following s
ystem of equations (a) by Gaussian elimination (b) by Gauss-Jordan eli
mination.  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEW
ISE([3*y-4*z = 1, ``],[9*x-4*y+z = 4, ``],[x+y+z = 15, ``])" "6#-%*PIE
CEWISEG6%7$/,&*&\"\"$\"\"\"%\"yGF+F+*&\"\"%F+%\"zGF+!\"\"F+%!G7$/,(*&
\"\"*F+%\"xGF+F+*&F.F+F,F+F0F/F+F.F17$/,(F7F+F,F+F/F+\"#:F1" }{TEXT 
-1 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 -1 102 "Solve the following
 system of equations (a) by Gaussian elimination (b) by Gauss-Jordan e
limination.  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC
EWISE([x+2*y-12*z = 2, ``],[4*x-y+z = 4, ``],[3*x+y+5*z = 11, ``])" "6
#-%*PIECEWISEG6%7$/,(%\"xG\"\"\"*&\"\"#F*%\"yGF*F**&\"#7F*%\"zGF*!\"\"
F,%!G7$/,(*&\"\"%F*F)F*F*F-F1F0F*F7F27$/,(*&\"\"$F*F)F*F*F-F**&\"\"&F*
F0F*F*\"#6F2" }{TEXT -1 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 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 
102 "Solve the following system of equations (a) by Gaussian eliminati
on (b) by Gauss-Jordan elimination.  " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "PIECEWISE([w+x+y = 3, ``],[-3*w-17*x+y+2*z = 1, ``
],[4*w-17*x+8*y-5*z = 1, ``],[-5*x-2*y+z = 1, ``])" "6#-%*PIECEWISEG6&
7$/,(%\"wG\"\"\"%\"xGF*%\"yGF*\"\"$%!G7$/,**&F-F*F)F*!\"\"*&\"#<F*F+F*
F3F,F**&\"\"#F*%\"zGF*F*F*F.7$/,**&\"\"%F*F)F*F**&F5F*F+F*F3*&\"\")F*F
,F*F**&\"\"&F*F8F*F3F*F.7$/,(*&FBF*F+F*F3*&F7F*F,F*F3F8F*F*F." }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "______________________________
_____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 116 "Find the inverse of
 the following matrix by Gauss-Jordan elimination, that is, by transfo
rming the identity matrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[2, 0, -1], [5, 1, 0], [0, 1, 3]])" "6#-%'matri
xG6#7%7%\"\"#\"\"!,$\"\"\"!\"\"7%\"\"&F+F)7%F)F+\"\"$" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 116 "Find the inverse of t
he following matrix by Gauss-Jordan elimination, that is, by transform
ing the identity matrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[1, 3, 2], [-2, -5, -1], [2, 4, 0]])" "6#-%'mat
rixG6#7%7%\"\"\"\"\"$\"\"#7%,$F*!\"\",$\"\"&F-,$F(F-7%F*\"\"%\"\"!" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "________________________
___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 116 "Find the inverse of
 the following matrix by Gauss-Jordan elimination, that is, by transfo
rming the identity matrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[1, 3, 4], [-1, -4, -2], [4, 9, 20]])" "6#-%'ma
trixG6#7%7%\"\"\"\"\"$\"\"%7%,$F(!\"\",$F*F-,$\"\"#F-7%F*\"\"*\"#?" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "________________________
___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 116 "Find the inverse of
 the following matrix by Gauss-Jordan elimination, that is, by transfo
rming the identity matrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "matrix([[2, 4, 0], [3, 4, -2], [-1, 1, 2]])" "6#-%'matr
ixG6#7%7%\"\"#\"\"%\"\"!7%\"\"$F),$F(!\"\"7%,$\"\"\"F.F1F(" }{TEXT -1 
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 "
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