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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,Jor
danBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(a
djointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatr
ixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspan
G%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG
%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eig
envaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponenti
alG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gauss
elimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(h
essianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)int
basisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'ke
rnelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(mi
npolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'or
thogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG
%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-sin
gularvalsG%&smithG%&stackG%*submatrixG%*subvectorG%)sumbasisG%(swapcol
G%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*
vecpotentG%(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 3 "" 0 "" 
{TEXT -1 17 "Matrix Operations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "Here is how to input a matrices A and B.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "A:=matrix(3,3,[1,2,1,0,-1,2,4,7,-6]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#F*7%\"\"!!\"\"F+7
%\"\"%\"\"(!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B:=matri
x(3,3,[4,2,1,-2,0,2,3,7,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG
-%'MATRIXG6#7%7%\"\"%\"\"#\"\"\"7%!\"#\"\"!F+7%\"\"$\"\"(F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 23 "Scalar multiply A by -2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "evalm(2*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7
%\"\"#\"\"%F(7%\"\"!!\"#F)7%\"\")\"#9!#7" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 48 "Note that without evalm, all you would get is 2A" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2*A;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%\"AG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Sc
alar division works too." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"evalm(A/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#\"\"
\"\"\"#F)F(7%\"\"!#!\"\"F*F)7%F*#\"\"(F*!\"$" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 21 "Add two matrices: A+B" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "evalm(A+B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MAT
RIXG6#7%7%\"\"&\"\"%\"\"#7%!\"#!\"\"F)7%\"\"(\"#9!\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "Multiply two matrices: A*B" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A&*B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"$\"\"*\"\"(7%\"\")\"#9\"\"#7%!#;!#M
\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The matrix A and then it
s transpose:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(A); t
ranspose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"
\"\"#F(7%\"\"!!\"\"F)7%\"\"%\"\"(!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'MATRIXG6#7%7%\"\"\"\"\"!\"\"%7%\"\"#!\"\"\"\"(7%F(F,!\"'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The inverse of A" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'MATRIXG6#7%7%#!\"#\"\"$#\"#>\"#7#\"\"&F-7%#\"\"#F*#!\"&\"\"'#
!\"\"F57%#\"\"\"F*#F:F-#F7F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "T
he negative power also works to find the inverse" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "evalm(A^(-1));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'MATRIXG6#7%7%#!\"#\"\"$#\"#>\"#7#\"\"&F-7%#\"\"#F*#!\"&\"\"'#
!\"\"F57%#\"\"\"F*#F:F-#F7F-" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 24 "S
olving Matrix Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 82 "Let's solve the equation Ax=b where b is the  column
 vector (1,4,-4) (transposed)." }}{PARA 0 "" 0 "" {TEXT -1 21 "First, \+
define A and b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix
(3,3,[1,2,1,0,-1,2,4,7,-6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG
-%'MATRIXG6#7%7%\"\"\"\"\"#F*7%\"\"!!\"\"F+7%\"\"%\"\"(!\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b:=matrix(3,1,[1,4,-4]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'MATRIXG6#7%7#\"\"\"7#\"\"%7#!
\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Then augment A with b (and
 call the result AA)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AA:
=augment(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG-%'MATRIXG6#7%
7&\"\"\"\"\"#F*F*7&\"\"!!\"\"F+\"\"%7&F/\"\"(!\"'!\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "Then apply Gaussian elimination" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "G:=gaussjord(AA);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'MATRIXG6#7%7&\"\"\"\"\"!F+\"\"%7&F+
F*F+!\"#7&F+F+F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The answer \+
is x=(4,-2,1) (transposed) - the last column of G. Let's check this." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x:=matrix(3,1,[4,-2,1]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'MATRIXG6#7%7#\"\"%7#!\"#7
#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A&*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7#\"\"\"7#\"\"%7#!\"%" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "which equals b. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Maple can extract th
e fourth column of G with the col command" }}{PARA 0 "" 0 "" {TEXT -1 
57 "so that you won't have to write out the solution by hand." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x:=col(G,4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"xG-%'VECTORG6#7%\"\"%!\"#\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Another w
ay to solve Ax=b is to solve for x as" }}{PARA 0 "" 0 "" {TEXT -1 48 "
x=inverse(A)*b. Note that the order is important" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "(b*inverse(A) would not make sense)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "x:=evalm(inverse(A)&*b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xG-%'MATRIXG6#7%7#\"\"%7#!\"#7#\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Here is another example" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix(3,3,[1,2,-1,0,1,-3,2,3,1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#!\"\"
7%\"\"!F*!\"$7%F+\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 
"b:=matrix(3,1,[-10,-13,-7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"b
G-%'MATRIXG6#7%7#!#57#!#87#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "AA:=augment(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG-%'
MATRIXG6#7%7&\"\"\"\"\"#!\"\"!#57&\"\"!F*!\"$!#87&F+\"\"$F*!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gaussjord(AA);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&\"\"\"\"\"!\"\"&\"#;7&F)F(!\"$!#
87&F)F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "There are an infin
ite number of solutions." }}{PARA 0 "" 0 "" {TEXT -1 44 "The third com
ponent can be arbitrary, x_3=c." }}{PARA 0 "" 0 "" {TEXT -1 30 "then x
_2=3c-13 and x_1=16-5c. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 39 "On the other hand, take the following b" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b:=matrix(3,1,[2,4,-5]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'MATRIXG6#7%7#\"\"#7#\"\"%7#!
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AA:=augment(A,b);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AAG-%'MATRIXG6#7%7&\"\"\"\"\"#!\"
\"F+7&\"\"!F*!\"$\"\"%7&F+\"\"$F*!\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "gaussjord(AA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
MATRIXG6#7%7&\"\"\"\"\"!\"\"&F)7&F)F(!\"$F)7&F)F)F)F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 27 "Here, there is no solution." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "C:=matrix(3,3,[1,
2,-1,0,0,0,2,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIX
G6#7%7%\"\"\"\"\"#!\"\"7%\"\"!F.F.7%F+\"\"$F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "E:=matrix(3,3,[1,0,0,2,1,-1,0,0,1]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%\"\"#F*
!\"\"7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(E&*C
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"#!\"\"
7%\"\"!F(!\"$7%F)\"\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 33 "Determinants and Other Operat
ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "D
eterminants. Here is a matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "A:=matrix(3,3,[1,2,1,0,-1,2,4,7,-6]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#F*7%\"\"!!\"\"F+7%\"\"
%\"\"(!\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Here is its determi
nant" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "He
re is a singular matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
A:=matrix(3,3,[1,2,-1,0,1,-3,2,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"#!\"\"7%\"\"!F*!\"$7%F+\"\"$F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Note that
 its inverse does not exist" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "inverse(A);" }}{PARA 8 "" 1 "" {TEXT -1 35 "Error, (in inverse) si
ngular matrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The adjoint of A
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix(3,3,[1,2,1,0,
-1,2,4,7,-6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%
7%\"\"\"\"\"#F*7%\"\"!!\"\"F+7%\"\"%\"\"(!\"'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "adjoint(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'MATRIXG6#7%7%!\")\"#>\"\"&7%\"\")!#5!\"#7%\"\"%\"\"\"!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Note that adjoint(A)/det(A) is the
 same as the inverse of A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "evalm(adjoint(A)/det(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MA
TRIXG6#7%7%#!\"#\"\"$#\"#>\"#7#\"\"&F-7%#\"\"#F*#!\"&\"\"'#!\"\"F57%#
\"\"\"F*#F:F-#F7F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inver
se(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#!\"#\"\"$#
\"#>\"#7#\"\"&F-7%#\"\"#F*#!\"&\"\"'#!\"\"F57%#\"\"\"F*#F:F-#F7F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 33 "Kernel, Row
space and Column Space" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:=matrix(3,5,[1,-1,0,2,3,2,0,2,2,8
,1,1,2,0,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7
'\"\"\"!\"\"\"\"!\"\"#\"\"$7'F-F,F-F-\"\")7'F*F*F-F,\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gaussjord(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F(F(\"\"%7'F)F(F(!\"\"F(7'F)
F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "NA:=kernel(A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NAG<%-%'VECTORG6#7'!\"\"F*\"\"
\"\"\"!F,-F'6#7'!\"%F*F,F,F+-F'6#7'F*F+F,F+F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "v:=matrix(5,1,op(2,NA));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"vG-%'MATRIXG6#7'7#!\"%7#!\"\"7#\"\"!F-7#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A&*v);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7#\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "gaussjord(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'MATRIXG6#7%7'\"\"\"\"\"!F(F(\"\"%7'F)F(F(!\"\"F(7'F)F)F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "rowspace(A)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7'\"\"\"\"\"!F(F(\"
\"%-F%6#7'F)F(F(!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
colspace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7%\"\"!
\"\"\"F)-F%6#7%F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "68 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }