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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 31 "Finding the inverse of \+
a matrix" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 205 127 "In the evening cl
ass we started finding out the inverse of the coefficient matrix in pr
oblem 13(a) section 1.3. The matrix was:" }}}{EXCHG {PARA 202 "> " 0 "
" {MPLTEXT 1 206 20 "with(LinearAlgebra);" }}}{EXCHG {PARA 202 "> " 0 
"" {MPLTEXT 1 206 39 "A:=Matrix([[2,-3,5],[9,-1,1],[1,5,4]]);" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 201 "" 
0 "" {TEXT 205 81 "We started the algorithm of finding the inverse by \+
defining the augmented matrix:" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 207 4 "A1:=" }{MPLTEXT 1 206 54 "Matrix([[2,-3,5,1,0,0],[9,
-1,1,0,1,0],[1,5,4,0,0,1]]);" }{MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 
201 "" 0 "" {TEXT 205 221 "The algorithm says that if we can bring A t
o its reduced row echelon form with three ones, then in the augmented \+
matrix from above we will get the inverse in the last three columns. W
e can do this directly by the command:" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 206 26 "ReducedRowEchelonForm(A1);" }}}{EXCHG {PARA 201 "" 
0 "" {TEXT 205 190 "But this is not the point. We want to \"record\" t
he row operations that need to be done in elementary matrices. So here
 are the steps. we first swap first and third row to get a pot in row \+
1." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 4 "E1:=" }{MPLTEXT 1 
207 34 "Matrix([[0,0,1],[0,1,0],[1,0,0]]);" }{MPLTEXT 1 206 0 "" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 20 "A2:=Multiply(E1,A1);" }}
}{EXCHG {PARA 201 "" 0 "" {TEXT 205 40 "Next we do Row2- 9*Row1 and Ro
w3-2*Row1:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 4 "E2:=" }
{MPLTEXT 1 207 35 "Matrix([[1,0,0],[-9,1,0],[0,0,1]]);" }{MPLTEXT 1 
206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 4 "A3:=" }
{MPLTEXT 1 206 16 "Multiply(E2,A2);" }{MPLTEXT 1 206 0 "" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 206 3 "E3:" }{MPLTEXT 1 206 1 "=" }
{MPLTEXT 1 207 35 "Matrix([[1,0,0],[0,1,0],[-2,0,1]]);" }{MPLTEXT 1 
206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 3 "A4:" }
{MPLTEXT 1 206 17 "=Multiply(E3,A3);" }{MPLTEXT 1 206 0 "" }}}{EXCHG 
{PARA 201 "" 0 "" {TEXT 205 145 "Now, since Maple can handle computati
ons much better than humans, we divide the second row by -46 and proce
ed to get zeros on the second column. " }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 206 4 "E4:=" }{MPLTEXT 1 207 38 "Matrix([[1,0,0],[0,-1/46,0
],[0,0,1]]);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 206 4 "A5:=" }{MPLTEXT 1 206 16 "Multiply(E4,A4);" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 4 "E5
:=" }{MPLTEXT 1 207 35 "Matrix([[1,0,0],[0,1,0],[0,13,1]]);" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 20 "A
6:=Multiply(E5,A5);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "
" {MPLTEXT 1 206 4 "E6:=" }{MPLTEXT 1 207 39 "Matrix([[1,0,0],[0,1,0],
[0,0,46/317]]);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 206 4 "A7:=" }{MPLTEXT 1 206 16 "Multiply(E6,A6);" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 4 "E7
:=" }{MPLTEXT 1 207 39 "Matrix([[1,0,0],[0,1,-35/46],[0,0,1]]);" }
{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 206 20 "A
8:=Multiply(E7,A7);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "
" {MPLTEXT 1 206 4 "E8:=" }{MPLTEXT 1 207 35 "Matrix([[1,0,-4],[0,1,0]
,[0,0,1]]);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 207 20 "A9:=Multiply(E8,A8);" }{MPLTEXT 1 206 0 "" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 207 39 "E9:=Matrix([[1,-5,0],[0,
1,0],[0,0,1]]);" }{MPLTEXT 1 206 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 207 21 "A10:=Multiply(E9,A9);" }{MPLTEXT 1 206 0 "" }}}
{EXCHG {PARA 201 "" 0 "" {TEXT 205 201 "And as you can see we obtained
 the reduced-row-echelon form of the matrix. Moreover, the matrix on t
he right represents he product of all elementary matrices E1 through E
9: E9*E8*E7*E6*E5*E4*E3*E2*E1. " }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }