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{SECT 0 {EXCHG {PARA 203 "" 0 "" {TEXT 207 43 "In-class Demonstration:
 Elementary Matrices" }{TEXT 207 0 "" }}{PARA 206 "" 0 "" {TEXT 201 0 
"" }}{PARA 206 "" 0 "" {TEXT 201 17 "By Russell Blyth." }{TEXT 201 0 "
" }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 0 29 "restart: with(LinearAlgebra):" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "randomize(3967);" }}{PARA 11
 "" 1 "" {XPPMATH 20 "\"%nR" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{SECT 0 {PARA 210 "" 0 "" {TEXT 208 36 "Elementary row and col
umn operations" }{TEXT 208 0 "" }}{EXCHG {PARA 206 "" 0 "" {TEXT 201 
31 "Generate a random 3 x 4 matrix." }{TEXT 201 0 "" }}}{EXCHG {PARA 
202 "> " 0 "" {MPLTEXT 1 0 45 "A := RandomMatrix(3,4,generator=rand(-5
..5));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"AG6\"-I
'RTABLEGF$6%\"('pMD-I'MATRIXGF$6#7%7&!#i\"\"&\"#N!#p7&!#K\"#t\"#W!#J7&
\"#$*\"\")!#5\"#%*I'MatrixG6$%*protectedGI(_syslibGF$" }}}{EXCHG 
{PARA 206 "" 0 "" {TEXT 201 62 "Also generate the 3x3 and 4x4 identity
 matrices for later use." }{TEXT 201 0 "" }}}{EXCHG {PARA 202 "> " 0 "
" {MPLTEXT 1 0 24 "I3 := IdentityMatrix(3);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 6 "I4 := " }{MPLTEXT 1 209 15 "Ident
ityMatrix(" }{MPLTEXT 1 0 3 "4);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I#I3G6\"-I'RTABLEGF$6%\"(7\"3D-I'MATRIXGF$6#7%7%\"\"\"
\"\"!F/7%F/F.F/7%F/F/F.I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I#I4G6\"-I'RTABLEGF$6%\"('zHD-I'MATRIXGF$6#7&7&
\"\"\"\"\"!F/F/7&F/F.F/F/7&F/F/F.F/7&F/F/F/F.I'MatrixG6$%*protectedGI(
_syslibGF$" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 51 "Apply several el
ementary row and column operations." }{TEXT 201 0 "" }}{PARA 206 "" 0 
"" {TEXT 201 124 "The three elementary row operations can be performed
 using the RowOperation command in the LinearAlgebra package of Maple.
  " }}{PARA 206 "" 0 "" {TEXT 201 12 "The command " }{TEXT 201 0 "" }}
{PARA 206 "" 0 "" {TEXT 201 32 "RowOperation(M, [r2, r1], scal);" }
{TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 57 "is used to add scal t
imes row r1 of matrix M to row r2.  " }}{PARA 206 "" 0 "" {TEXT 201 
11 "The command" }{TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 12 "Row
Operation" }{TEXT 201 14 "(M, r1, scal);" }{TEXT 201 0 "" }}{PARA 206 
"" 0 "" {TEXT 201 54 "is used to multiply row r1 of M by scal.  The co
mmand " }{TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 12 "RowOperation
" }{TEXT 201 14 "(M, [r1, r2]);" }{TEXT 201 0 "" }}{PARA 206 "" 0 "" 
{TEXT 201 49 "is used to switch rows r1 and r2 of the matrix M." }
{TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 206 "" 0 "" 
{TEXT 201 57 "ColumnOperation() performs similar operations on columns
." }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 
306 "We apply several elementary row operations to the matrix A; at th
e same time we compute the corresponding elementary matrices by applyi
ng the same row operations to the identity matrix. We can then check t
hat multiplying A on the left by the elementary matrix produces the sa
me result as the row operation." }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 0 22 "RowOperation(A,[1,2]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 
1 0 2 "\n" }{MPLTEXT 1 0 6 "E1 := " }{MPLTEXT 1 209 12 "RowOperation" 
}{MPLTEXT 1 0 11 "(I3,[1,2]);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n"
 }{MPLTEXT 1 0 5 "E1.A;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "-I'RTABLEG6\"6%\"(w?d#-I'MATRIXGF$6#7%7&\"\"&!\"#\"\"!!\"
\"7&\"\"\"\"\"%F.!\"%7&F,!\"&\"\"#F/I'MatrixG6$%*protectedGI(_syslibGF
$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E1G6\"-I'RTABLEGF$6%\"(oDf#-I'MA
TRIXGF$6#7%7%\"\"!\"\"\"F.7%F/F.F.7%F.F.F/I'MatrixG6$%*protectedGI(_sy
slibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(g&>F-I'MATR
IXGF$6#7%7&\"\"&!\"#\"\"!!\"\"7&\"\"\"\"\"%F.!\"%7&F,!\"&\"\"#F/I'Matr
ixG6$%*protectedGI(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 
1 209 12 "RowOperation" }{MPLTEXT 1 0 10 "(A,3,1/3);" }{MPLTEXT 1 0 0 
"" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 6 "E2 := " }{MPLTEXT 1 209 12 "R
owOperation" }{MPLTEXT 1 0 11 "(I3,3,1/3);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 5 "E2.A;" }{MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(G'\\F-I'MATRIXGF$6#7%7
&\"\"\"\"\"%\"\"!!\"%7&\"\"&!\"#F.!\"\"7&#F1\"\"$#!\"&F6#\"\"#F6#F3F6I
'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
#E2G6\"-I'RTABLEGF$6%\"(/Ix#-I'MATRIXGF$6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F
/F/#F.\"\"$I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "-I'RTABLEG6\"6%\"(o*4G-I'MATRIXGF$6#7%7&\"\"\"\"\"%\"\"!!
\"%7&\"\"&!\"#F.!\"\"7&#F1\"\"$#!\"&F6#\"\"#F6#F3F6I'MatrixG6$%*protec
tedGI(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 209 12 "Row
Operation" }{MPLTEXT 1 0 13 "(A,[3,1],-1);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 6 "E3 := " }{MPLTEXT 1 209 12 "RowOp
eration" }{MPLTEXT 1 0 14 "(I3,[3,1],-1);" }{MPLTEXT 1 0 0 "" }
{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 5 "E3.A;" }{MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(;g$G-I'MATRIXGF$6#7%7&
\"\"\"\"\"%\"\"!!\"%7&\"\"&!\"#F.!\"\"7&F-!\"*\"\"#\"\"$I'MatrixG6$%*p
rotectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E3G6\"-I'RTA
BLEGF$6%\"(W\\&G-I'MATRIXGF$6#7%7%\"\"\"\"\"!F/7%F/F.F/7%!\"\"F/F.I'Ma
trixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RT
ABLEG6\"6%\"(sf*G-I'MATRIXGF$6#7%7&\"\"\"\"\"%\"\"!!\"%7&\"\"&!\"#F.!
\"\"7&F-!\"*\"\"#\"\"$I'MatrixG6$%*protectedGI(_syslibGF$" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 0 25 "ColumnOperation(A,[2,4]);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 6 "E4 := " }
{MPLTEXT 1 209 15 "ColumnOperation" }{MPLTEXT 1 0 11 "(I4,[2,4]);" }
{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 5 "A.E4;" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(35
#H-I'MATRIXGF$6#7%7&\"\"\"!\"%\"\"!\"\"%7&\"\"&!\"\"F.!\"#7&F1F2\"\"#!
\"&I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I#E4G6\"-I'RTABLEGF$6%\"(g;%H-I'MATRIXGF$6#7&7&\"\"\"\"\"!F/F/7&F/F
/F/F.7&F/F/F.F/7&F/F.F/F/I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(C&yH-I'MATRIXGF$6#7%7&\"\"\"
!\"%\"\"!\"\"%7&\"\"&!\"\"F.!\"#7&F1F2\"\"#!\"&I'MatrixG6$%*protectedG
I(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 209 15 "ColumnO
peration" }{MPLTEXT 1 0 8 "(A,1,4);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 
2 "\n" }{MPLTEXT 1 0 6 "E5 := " }{MPLTEXT 1 209 15 "ColumnOperation" }
{MPLTEXT 1 0 9 "(I4,1,4);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 5 "A.E5;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "-I'RTABLEG6\"6%\"(3E+$-I'MATRIXGF$6#7%7&\"\"%F,\"\"!!\"%7&\"#?!\"#
F-!\"\"7&F0!\"&\"\"#F2I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I#E5G6\"-I'RTABLEGF$6%\"(Gh-$-I'MATRIXGF$6#7&7&
\"\"%\"\"!F/F/7&F/\"\"\"F/F/7&F/F/F1F/7&F/F/F/F1I'MatrixG6$%*protected
GI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(W22$-
I'MATRIXGF$6#7%7&\"\"%F,\"\"!!\"%7&\"#?!\"#F-!\"\"7&F0!\"&\"\"#F2I'Mat
rixG6$%*protectedGI(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT
 1 209 15 "ColumnOperation" }{MPLTEXT 1 0 13 "(A,[4,1],-2);" }{MPLTEXT
 1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 6 "E6 := " }{MPLTEXT 1 
209 15 "ColumnOperation" }{MPLTEXT 1 0 14 "(I4,[4,1],-2);" }{MPLTEXT 
1 0 0 "" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 5 "A.E6;" }{MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(;H4$-I'MATRIXGF$6
#7%7&\"\"\"\"\"%\"\"!!\"'7&\"\"&!\"#F.!#67&F1!\"&\"\"#F3I'MatrixG6$%*p
rotectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#E6G6\"-I'RTA
BLEGF$6%\"(3AB%-I'MATRIXGF$6#7&7&\"\"\"\"\"!F/!\"#7&F/F.F/F/7&F/F/F.F/
7&F/F/F/F.I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "-I'RTABLEG6\"6%\"(oAF%-I'MATRIXGF$6#7%7&\"\"\"\"\"%\"\"!!
\"'7&\"\"&!\"#F.!#67&F1!\"&\"\"#F3I'MatrixG6$%*protectedGI(_syslibGF$"
 }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 200 "Note that multiplication b
y an elementary matrix performs the corresponding elementary row or co
lumn operation. For a row operation the elementary matrix is multiplie
d on the left, while for a column " }{TEXT 201 58 "operation the eleme
ntary matrix is multiplied on the right" }{TEXT 201 1 "." }{TEXT 201 
0 "" }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 
71 "Now check the inverses of each of the elementary matrices we gener
ated." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 12 "E1; E1^(-1);" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(oD
f#-I'MATRIXGF$6#7%7%\"\"!\"\"\"F,7%F-F,F,7%F,F,F-I'MatrixG6$%*protecte
dGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(s4O%
-I'MATRIXGF$6#7%7%\"\"!\"\"\"F,7%F-F,F,7%F,F,F-I'MatrixG6$%*protectedG
I(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 12 "E2; E2^(-
1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6
%\"(/Ix#-I'MATRIXGF$6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F-#F,\"\"$I'MatrixG
6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG
6\"6%\"(WPS%-I'MATRIXGF$6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F-\"\"$I'Matrix
G6$%*protectedGI(_syslibGF$" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 
0 12 "E3; E3^(-1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 
"-I'RTABLEG6\"6%\"(W\\&G-I'MATRIXGF$6#7%7%\"\"\"\"\"!F-7%F-F,F-7%!\"\"
F-F,I'MatrixG6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 
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{PARA 11 "" 1 "" {XPPMATH 20 "-I'RTABLEG6\"6%\"(w8p%-I'MATRIXGF$6#7&7&
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trixG6$%*protectedGI(_syslibGF$" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 
203 252 "  Note that each elementary matrix is invertible, and that it
's inverse is the elementary matrix corresponding to the elementary ro
w or column operation that undoes the elementary row or column operati
on corresponding to the original elementary matrix." }}}{EXCHG {PARA 
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