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{SECT 0 {EXCHG {PARA 201 "" 0 "" {TEXT 204 43 "In-class Demonstration:
 Elementary Matrices" }{TEXT 204 0 "" }}{PARA 202 "" 0 "" {TEXT 205 0 
"" }}{PARA 202 "" 0 "" {TEXT 205 17 "By Russell Blyth." }{TEXT 205 0 "
" }}{PARA 202 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 0 29 "restart: with(LinearAlgebra):" }{MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 204 "" 0 "" {TEXT 206 36 "Elementary row and column oper
ations" }{TEXT 206 0 "" }}{EXCHG {PARA 202 "" 0 "" {TEXT 205 31 "Gener
ate a random 3 x 4 matrix." }{TEXT 205 0 "" }}}{EXCHG {PARA 203 "> " 0
 "" {MPLTEXT 1 0 45 "A := RandomMatrix(3,4,generator=rand(-5..5));" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 205 62 "Also gener
ate the 3x3 and 4x4 identity matrices for later use." }{TEXT 205 0 "" 
}}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 24 "I3 := IdentityMatrix(3)
;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nI4 := " }{MPLTEXT 1 -1 15 "Id
entityMatrix(" }{MPLTEXT 1 0 3 "4);" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 202 "" 0 "" {TEXT 205 51 "Apply several elementary row and colum
n operations." }{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 124 "The \+
three elementary row operations can be performed using the RowOperatio
n command in the LinearAlgebra package of Maple.  " }}{PARA 202 "" 0 "
" {TEXT 205 12 "The command " }{TEXT 205 0 "" }}{PARA 202 "" 0 "" 
{TEXT 205 32 "RowOperation(M, [r2, r1], scal);" }{TEXT 205 0 "" }}
{PARA 202 "" 0 "" {TEXT 205 57 "is used to add scal times row r1 of ma
trix M to row r2.  " }}{PARA 202 "" 0 "" {TEXT 205 11 "The command" }
{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 12 "RowOperation" }{TEXT 
205 14 "(M, r1, scal);" }{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 
54 "is used to multiply row r1 of M by scal.  The command " }{TEXT 
205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 12 "RowOperation" }{TEXT 205 
14 "(M, [r1, r2]);" }{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 49 "
is used to switch rows r1 and r2 of the matrix M." }{TEXT 205 0 "" }}
{PARA 202 "" 0 "" {TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 57 "Col
umnOperation() performs similar operations on columns." }}{PARA 202 ""
 0 "" {TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 306 "We apply sever
al elementary row operations to the matrix A; at the same time we comp
ute the corresponding elementary matrices by applying the same row ope
rations to the identity matrix. We can then check that multiplying A o
n the left by the elementary matrix produces the same result as the ro
w operation." }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 22 "RowOperat
ion(A,[1,2]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE1 := " }
{MPLTEXT 1 -1 12 "RowOperation" }{MPLTEXT 1 0 11 "(I3,[1,2]);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 6 "\nE1.A;" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 -1 12 "RowOperation" }{MPLTEXT 
1 0 10 "(A,3,1/3);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE2 := " }
{MPLTEXT 1 -1 12 "RowOperation" }{MPLTEXT 1 0 11 "(I3,3,1/3);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 6 "\nE2.A;" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 -1 12 "RowOperation" }{MPLTEXT 
1 0 13 "(A,[3,1],-1);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE3 := " }
{MPLTEXT 1 -1 12 "RowOperation" }{MPLTEXT 1 0 14 "(I3,[3,1],-1);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 6 "\nE3.A;" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 25 "ColumnOperation(A,[2,4]);"
 }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE4 := " }{MPLTEXT 1 -1 15 "Colu
mnOperation" }{MPLTEXT 1 0 11 "(I4,[2,4]);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 6 "\nA.E4;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0
 "" {MPLTEXT 1 -1 15 "ColumnOperation" }{MPLTEXT 1 0 8 "(A,1,4);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE5 := " }{MPLTEXT 1 -1 15 "Column
Operation" }{MPLTEXT 1 0 9 "(I4,1,4);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 
0 6 "\nA.E5;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 -1 15 "ColumnOperation" }{MPLTEXT 1 0 13 "(A,[4,1],-2);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 7 "\nE6 := " }{MPLTEXT 1 -1 15 "Column
Operation" }{MPLTEXT 1 0 14 "(I4,[4,1],-2);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 6 "\nA.E6;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "" 0 "
" {TEXT 205 200 "Note that multiplication by an elementary matrix perf
orms the corresponding elementary row or column operation. For a row o
peration the elementary matrix is multiplied on the left, while for a \+
column " }{TEXT 205 58 "operation the elementary matrix is multiplied \+
on the right" }{TEXT 205 1 "." }{TEXT 205 0 "" }}{PARA 202 "" 0 "" 
{TEXT 205 0 "" }}{PARA 202 "" 0 "" {TEXT 205 71 "Now check the inverse
s of each of the elementary matrices we generated." }}}{EXCHG {PARA 
203 "> " 0 "" {MPLTEXT 1 0 12 "E1; E1^(-1);" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 12 "E2; E2^(-1);" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 12 "E3; E3^(-1);" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 12 "E4; E
4^(-1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 
0 12 "E5; E5^(-1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 0 12 "E6; E6^(-1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 205 
"" 0 "" {TEXT 207 252 "  Note that each elementary matrix is invertibl
e, and that it's inverse is the elementary matrix corresponding to the
 elementary row or column operation that undoes the elementary row or \+
column operation corresponding to the original elementary matrix." }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 206 "" 0 "" 
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