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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 258 24 "Linear A
lgebra Powertool" }}{PARA 256 "" 0 "" {TEXT 256 31 "Finding the Invers
e of a Matrix" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 26 "Workshee
t by Russell Blyth" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Enter matrix
 to be inverted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=matri
x([[1,4,-1],[-2,0,3],[3,6,1]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Augment the matrix by \+
appending the identity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "K
:=augment(A,[1,0,0],[0,1,0],[0,0,1]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Reduce the au
gmented matrix to reduced row echelon form" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "K:=rref(K);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "If the n x n identity
 appears in the first n columns of the matrix, the last n x n columns \+
is the inverse; extract it using the submatrix command" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "AINV:=submatrix(K,1..3,4..6);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "Check the result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "evalm(A &* AINV);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "evalm(AINV &* A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The linalg package has a built-i
n inverse command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "invers
e(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 157 "If the matrix has no inverse, the procedure ab
ove will not give the nx n identity in the first n x n columns of K. H
ere is how the inverse function behaves: " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "B:=matrix([[2,4],[4,8]]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "in
verse(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "The linalg package \+
includes a routine to generate random matrices with integer entries, f
or testing purposes. Let's use it to generate matrices and inverses." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C:=randmatrix(4,4);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "inverse(C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "By default, the entri
es of the randomly generated matrix fall between -99 and 99. An option
 can be added to specify the allowed range." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "E:=randmatrix(4,4,entries=rand(-5..5));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 77 "(The variable name D is protected, and is used for the di
fferential operator)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(x^
2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(E);" }}}
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