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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 50 "Invertibility of A Matrix and S
ystems of Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 81 "Linear Algebra Project                                 \+
                          " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 12 " Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 29 "Use 'linalg
' package of Maple" }}{PARA 0 "" 0 "" {TEXT -1 48 "1.  To find the inv
erse of a given square matrix" }}{PARA 0 "" 0 "" {TEXT -1 50 "2.  To u
se the inverse to solve the system Ax = b " }}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 0 "" }{TEXT 258 22 " Required Information:" }}{PARA 0 "" 0 
"" {TEXT -1 53 " A n x n square matrix A is invertible if and only if
" }}{PARA 0 "" 0 "" {TEXT -1 61 "   the reduced row echelon form of A \+
is the identity matrix  " }{XPPEDIT 18 0 "I[n];" "6#&%\"IG6#%\"nG" }
{TEXT -1 17 "  i.e. rref(A) = " }{XPPEDIT 18 0 "I[n];" "6#&%\"IG6#%\"n
G" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We can only find inv
erses of square matrices.  A matrix which has no inverse is called a s
ingular matrix.  An invertible matrix is said to be nonsingular." }}
{PARA 0 "" 0 "" {TEXT -1 74 "If A is invertible then the solution of t
he system Ax = b is found as x = " }{XPPEDIT 18 0 "A^(-1);" "6#)%\"AG,
$\"\"\"!\"\"" }{TEXT -1 1 "b" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 257 18 " Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 58 "Fi
nd the inverses of the following matrices if they exist." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "A = " }{XPPEDIT 18 
0 "matrix([[1, 2, 3], [5, 6, 7], [6, 7, 8]]);" "6#-%'matrixG6#7%7%\"\"
\"\"\"#\"\"$7%\"\"&\"\"'\"\"(7%F-F.\"\")" }{TEXT -1 21 ",             \+
   B = " }{XPPEDIT 18 0 "matrix([[1, 2, 3], [2, 5, 3], [1, 0, 8]]);" "
6#-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%F)\"\"&F*7%F(\"\"!\"\")" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 257 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 10 " Solution:" }}{PARA 0 "" 0 "" {TEXT -1 
368 "To find the inverse of a matrix A, we find a squence of elementar
y operations that reduces A to the identity matrix and simlutaneously \+
perform the same sequence of operations on I to obtain A-inverse.  In \+
doing so, if A cannot be reduced to the identity matrix then the inver
se does not exist.  Thus augmenting the matrix A with I and trying to \+
reduce the matrix A to " }{TEXT 260 2 "I " }{TEXT -1 117 " will not on
ly show us if the matrix is invertible, but side by side will also pro
duce the inverse of A if it exists." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 89 "Finding an inverse of a matrix can also \+
be done in one step using the 'inverse' command,." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg)
:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm an
d trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 73 "A := matrix(3,3,[1,2,3,5,6,7,6,7,8]);I3:=matrix(
3,3,[1,0,0,0,1,0,0,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'
matrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"&\"\"'\"\"(7%F/F0\"\")" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#I3G-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+
F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "AI := augment(A,I3)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AIG-%'matrixG6#7%7(\"\"\"\"\"#
\"\"$F*\"\"!F-7(\"\"&\"\"'\"\"(F-F*F-7(F0F1\"\")F-F-F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "AI1 :=addrow(AI,1,2,-5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$AI1G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"
\"!F-7(F-!\"%!\")!\"&F*F-7(\"\"'\"\"(\"\")F-F-F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "AI2 := mulrow(AI1, 2, -1/4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$AI2G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(
F-F*F+#\"\"&\"\"%#!\"\"F1F-7(\"\"'\"\"(\"\")F-F-F*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "AI3 := addrow(AI2, 1, 3,-6);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$AI3G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-
7(F-F*F+#\"\"&\"\"%#!\"\"F1F-7(F-!\"&!#5!\"'F-F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "AI4 := addrow(AI3,2,3,5);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$AI4G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F
*F+#\"\"&\"\"%#!\"\"F1F-7(F-F-F-#F*F1#!\"&F1F*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 92 "We can stop at this stage since in the matrix AI4 th
e last row has three entries 0;  but in " }{XPPEDIT 18 0 "I[3];" "6#&%
\"IG6#\"\"$" }{TEXT -1 32 " the last row has third entry 1." }}{PARA 
0 "" 0 "" {TEXT -1 37 "Hence the matrix A is not invertible." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Using the 'inve
rse' command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ainverse= \+
inverse(A);" }}{PARA 8 "" 1 "" {TEXT -1 36 "Error, (in inverse) singul
ar matrix\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B:= matrix(3
,3,[1,2,3,2,5,3,1,0,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'m
atrixG6#7%7%\"\"\"\"\"#\"\"$7%F+\"\"&F,7%F*\"\"!\"\")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "BI := augment(B, I3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#BIG-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7
(F+\"\"&F,F-F*F-7(F*F-\"\")F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "BI1:= addrow(BI,1,2,-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$BI1G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F*!
\"$!\"#F*F-7(F*F-\"\")F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "BI2:= addrow(BI1,1,3,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
BI2G-%'matrixG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F*!\"$!\"#F*F-7(F-F0
\"\"&!\"\"F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "BI3:= add
row(BI2, 2,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BI3G-%'matrixG6
#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-F*!\"$!\"#F*F-7(F-F-!\"\"!\"&F+F*" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "BI4:= mulrow(BI3, 3, -1);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BI4G-%'matrixG6#7%7(\"\"\"\"\"#
\"\"$F*\"\"!F-7(F-F*!\"$!\"#F*F-7(F-F-F*\"\"&F0!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "BI5:= addrow(BI4,2,1,-2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$BI5G-%'matrixG6#7%7(\"\"\"\"\"!\"\"*\"\"&!\"#F+
7(F+F*!\"$F.F*F+7(F+F+F*F-F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "BI6:= addrow(BI5, 3,2,3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$BI6G-%'matrixG6#7%7(\"\"\"\"\"!\"\"*\"\"&!\"#F+7(F+F
*F+\"#8!\"&!\"$7(F+F+F*F-F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "BI7:= addrow(BI6, 3,1,-9);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$BI7G-%'matrixG6#7%7(\"\"\"\"\"!F+!#S\"#;\"\"*7(F+F*F
+\"#8!\"&!\"$7(F+F+F*\"\"&!\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "B^(-1);" "6#)%\"BG,$\"\"\"!\"\"" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "matrix([[-40, 16, 9], [13, -5, -3], [5, -2, -1]]);" "6#
-%'matrixG6#7%7%,$\"#S!\"\"\"#;\"\"*7%\"#8,$\"\"&F*,$\"\"$F*7%F0,$\"\"
#F*,$\"\"\"F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "B^(-1);" "6#)%\"BG,$\"\"\"!\"\"" }
{TEXT -1 25 " using 'inverse'. comand." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(B);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!#S\"#;\"\"*7%\"#8!\"&!\"$7%\"
\"&!\"#!\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 18 
" Solved Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 64 "Solve the followin
g linear system Ax = b using matrix inversion." }}{PARA 0 "" 0 "" 
{TEXT -1 17 "  x + 2y + 3z = 5" }}{PARA 0 "" 0 "" {TEXT -1 17 "2x + 5y
 + 3 z = 3" }}{PARA 0 "" 0 "" {TEXT -1 21 "x +           8z = 17" }}
{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "A^(-1);" "6#)%\"AG,
$\"\"\"!\"\"" }{TEXT -1 24 " using a single command." }}{PARA 0 "" 0 "
" {TEXT -1 9 "Find x = " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"
\"" }{TEXT -1 16 "b as a solution." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }{TEXT 262 0 "" }{TEXT 263 10 " Solution:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A:= matrix(3
,3,[1,2,3,2,5,3,1,0,8]); b:= matrix(3,1,[5, 3, 17]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%F+\"\"&F,7%F
*\"\"!\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'matrixG6#7%7#
\"\"&7#\"\"$7#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Ainve
rse := inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)AinverseG-%'m
atrixG6#7%7%!#S\"#;\"\"*7%\"#8!\"&!\"$7%\"\"&!\"#!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:= evalm(Ainverse&*(b));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'matrixG6#7%7#\"\"\"7#!\"\"7#\"\"#" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "This can also be obtained by u
sing multiply command as follows.  But 'evalm' command is more general
 and can evaluate a matrix expression involving *, +, -, power, and pa
rentheses.  Notice above the syntax of evalm in multiplying." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "x1 := multiply(Ainverse, b);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#x1G-%'matrixG6#7%7#\"\"\"7#!\"\"7#\"\"#" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "By both commands we get the same answer.
" }}}{PARA 0 "" 0 "" {TEXT -1 67 "____________________________________
_______________________________" }}{PARA 256 "" 0 "" {TEXT -1 10 "ASSI
GNMENT" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 264 11 " Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 
57 "Find the inverse of the following matrices if they exist." }}
{PARA 0 "" 0 "" {TEXT -1 87 "a) by augmenting the matrix with the iden
tity matrix and reducing to rref step by step." }}{PARA 0 "" 0 "" 
{TEXT -1 46 "b) by using the 'inverse' command in one step." }}{PARA 
0 "" 0 "" {TEXT -1 3 "A =" }{XPPEDIT 18 0 "matrix([[-1, 3, -4], [2, 4,
 1], [-4, 2, -9]]);" "6#-%'matrixG6#7%7%,$\"\"\"!\"\"\"\"$,$\"\"%F*7%
\"\"#F-F)7%,$F-F*F/,$\"\"*F*" }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "B = " }{XPPEDIT 18 0 "matr
ix([[1, 0, 0, 0], [1, 3, 0, 0], [2, 6, 10, 0], [3, 9, 15, 21]]);" "6#-
%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F(\"\"$F)F)7&\"\"#\"\"'\"#5F)7&F+\"\"*
\"#:\"#@" }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 265 11 " Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 24 "Solve the sy
stem Ax = b " }}{PARA 0 "" 0 "" {TEXT -1 15 "by finding the " }
{XPPEDIT 18 0 "A^(-1);" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 35 "in singl
e command, and finding x = " }{XPPEDIT 18 0 "A^(-1);" "6#)%\"AG,$\"\"
\"!\"\"" }{TEXT -1 1 "b" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 26 "    2w -   x + 3y + 2z = 3" }}{PARA 0 "" 0 "" {TEXT 
-1 24 "-2w +   x + 5y +   z = 4" }}{PARA 0 "" 0 "" {TEXT -1 22 "-3w + \+
2x + 2y - 3z = 1" }}{PARA 0 "" 0 "" {TEXT -1 24 " 4w  - 3x +   y + 3z \+
= 2" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 258 "" 0 "" {TEXT -1 
77 "__________________________________________________________________
___________" }}{PARA 0 "" 0 "" {TEXT -1 172 "MSIP Grant #P120A80089-98
:  \"Three Urban Calculus Reform programs: Adopting the Best\" 1998-20
01, MSEIP Grant #P120A010031: \"Four Colleges: Calculus + Enhancements
\"  2001-04" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2 0" 24 }
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