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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "with(LinearAlgebra);
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{PARA 12 "" 1 "" {XPPMATH 20 "6#7br%#&xG%$AddG%(AdjointG%3BackwardSubs
tituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-Bilinear
FormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0Colu
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2GetResultDataTypeG%/GetResultShapeG%5GivensRotationMatrixG%,GramSchmi
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0 "> " 0 "" {MPLTEXT 1 0 14 "?LinearAlgebra" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 106 "# the line above will show the help page for the \+
package (try ? in front of any word to get relevant help)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A:=Matrix([[1,3],[2,4]]);  #example
 of setting up a matrix." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'R
TABLEG6%\"(G)*R&-%'MATRIXG6#7$7$\"\"\"\"\"$7$\"\"#\"\"%%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"(G)*R&-%'MATRIXG6#7$7$\"\"\"\"\"$7$\"\"#
\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "MatrixInv
erse(A);A^(-1);  #two notations for matrix inverse." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")
wZjL-%'MATRIXG6#7$7$!\"##\"\"$\"\"#7$\"\"\"#!\"\"F/%'MatrixG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")#RkO$-%'MATRIXG6#7$7$!\"##\"
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{MPLTEXT 1 0 25 "B:=Matrix([[3,1],[2,4]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"(scd&-%'MATRIXG6#7$7$\"\"$\"\"\"7$
\"\"#\"\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A.B;
  #notation for matrix product; * represents multiplication of numbers
." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(OcE&-%'MATRIXG6#7$
7$\"\"*\"#87$\"#9\"#=%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "C:=Matrix([[1,2,3],[-1,0,2],[3,3,1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\"(?UO&-%'MATRIXG6#7%7%\"\"\"\"\"#\"
\"$7%!\"\"\"\"!F/7%F0F0F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "MatrixInverse(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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\"$!\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "?RowOpe
ration;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "E:=Matrix([C,Id
entityMatrix(3)]);  #notice that a list of matrices will be put togeth
er in a sensible way by the Matrix command." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"EG-%'RTABLEG6%\")/&*yL-%'MATRIXG6#7%7(\"\"\"\"\"#\"
\"$F.\"\"!F17(!\"\"F1F/F1F.F17(F0F0F.F1F1F.%'MatrixG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "rop:=RowOperation;  #abbreviate the RowOp
eration command." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ropG%-RowOperat
ionG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "#the following func
tion constructs elementary matrices." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "em:=(n,x,y)->if y=0 then rop(IdentityMatrix(n),x) els
e rop(IdentityMatrix(n),x,y) end if;" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#>%#emGf*6%%\"nG%\"xG%\"yG6\"6$%)operatorG%&arrowGF*@%/9&\"\"!-%$ropG
6$-_%.LinearAlgebraG%/IdentityMatrixG6#9$9%-F36%F5F;F0F*F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "#the following development s
hows how to use Maple to find the inverse of E using row operations." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "# I put in extra steps to
 find elementary matrices for each row operation; this is why I" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "# use %% instead of % -- th
is looks for the next-to-last result of a calculation instead of the l
ast result (if I used % I would apply the row operation to the element
ary matrix from the previous step, which is not what I want)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "E;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")/&*yL-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F,
\"\"!F/7(!\"\"F/F-F/F,F/7(F.F.F,F/F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "rop(%,[2,1],1);  E1:=em(3,[2,1],1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/!*4N-%'MATRIXG6#7%7(\"\"\"\"
\"#\"\"$F,\"\"!F/7(F/F-\"\"&F,F,F/7(F.F.F,F/F/F,%'MatrixG" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#E1G-%'RTABLEG6%\"(!=Oc-%'MATRIXG6#7%7%\"\"\"
\"\"!F/7%F.F.F/7%F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "rop(%%,[3,1],-3); E2:=em(3,[3,1],-3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")7S\\K-%'MATRIXG6#7%7(\"\"\"\"\"#\"
\"$F,\"\"!F/7(F/F-\"\"&F,F,F/7(F/!\"$!\")F3F/F,%'MatrixG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#E2G-%'RTABLEG6%\")GU<N-%'MATRIXG6#7%7%\"\"\"
\"\"!F/7%F/F.F/7%!\"$F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "rop(%%,2,1/2);  E3:=em(3,2,1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"(/y%e-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F,\"
\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/!\"$!\")F5F/F,%'MatrixG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#E3G-%'RTABLEG6%\")k/AN-%'MATRIXG6#7%7%\"\"\"
\"\"!F/7%F/#F.\"\"#F/7%F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "rop(%%,[3,2],3); E4:=em(3,[3,2],3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'RTABLEG6%\"(K9l&-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F
,\"\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/F/#!\"\"F-#!\"$F-#F.F-F,%'MatrixG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E4G-%'RTABLEG6%\")?nEN-%'MATRIXG6
#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/\"\"$F.%'MatrixG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "rop(%%,3,-2); E5:=em(3,3,-2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(#R$p&-%'MATRIXG6#7%7(\"\"\"\"\"#\"
\"$F,\"\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/F/F,F.!\"$!\"#%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E5G-%'RTABLEG6%\")!)RBK-%'MATRIXG6#
7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/!\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "rop(%%,[2,3],-5/2); E6:=em(3,[2,3],-5/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"))o=D$-%'MATRIXG6#7%7(\"
\"\"\"\"#\"\"$F,\"\"!F/7(F/F,F/!\"(\"\")\"\"&7(F/F/F,F.!\"$!\"#%'Matri
xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E6G-%'RTABLEG6%\")#\\e`$-%'MA
TRIXG6#7%7%\"\"\"\"\"!F/7%F/F.#!\"&\"\"#7%F/F/F.%'MatrixG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rop(%%,[1,3],-3); E7:=em(3,[1,3],-3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Gs9L-%'MATRIXG6#7
%7(\"\"\"\"\"#\"\"!!\")\"\"*\"\"'7(F.F,F.!\"(\"\")\"\"&7(F.F.F,\"\"$!
\"$!\"#%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E7G-%'RTABLEG6%
\"))[-a$-%'MATRIXG6#7%7%\"\"\"\"\"!!\"$7%F/F.F/7%F/F/F.%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rop(%%,[1,2],-2); E8:=em(3,[
1,2],-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")_QUN-%'MAT
RIXG6#7%7(\"\"\"\"\"!F-\"\"'!\"(!\"%7(F-F,F-F/\"\")\"\"&7(F-F-F,\"\"$!
\"$!\"#%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E8G-%'RTABLEG6%
\"(Cb%e-%'MATRIXG6#7%7%\"\"\"!\"#\"\"!7%F0F.F07%F0F0F.%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MatrixInverse(C);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+aGK-%'MATRIXG6#7%7%\"\"'!\"(
!\"%7%F-\"\")\"\"&7%\"\"$!\"$!\"#%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")s)[c$-%'MATRIXG6#7%7%\"\"'!\"(!\"%7%F-\"
\")\"\"&7%\"\"$!\"$!\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 176 "E8.E7.E6.E5.E4.E3.E2.E1;  #notice that the inverse i
s the product of the elementary matrices I constructed.  How could you
 express C itself as a product of elementary matrices?" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")SS-O-%'MATRIXG6#7%7%\"\"'!\"(!\"%7%
F-\"\")\"\"&7%\"\"$!\"$!\"#%'MatrixG" }}}{PARA 11 "" 1 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "C.%;  #the product of C \+
with this matrix is the identity as expected." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")k]FK-%'MATRIXG6#7%7%\"\"\"\"\"!F-7%F-F,F
-7%F-F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "F:=Ma
trix([[1,2,3],[3,4,5]]);  #a non-square matrix" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"FG-%'RTABLEG6%\"(;Gn&-%'MATRIXG6#7$7%\"\"\"\"\"#\"
\"$7%F0\"\"%\"\"&%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "C.F;  #an illegal multiplication" }}{PARA 8 "" 1 "" {TEXT -1 117 "
Error, (in LinearAlgebra:-MatrixMatrixMultiply) first matrix column di
mension (3) <> second matrix row dimension (2)\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "F.C;  #a legal multiplication" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(!Q)G&-%'MATRIXG6#7$7%\"\")\"#6\"#57
%\"#9\"#@\"#A%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "
MatrixInverse(F);  #Maple does something unexpected!  We don't define \+
inverses for non-square matrices." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'RTABLEG6%\")_A'f$-%'MATRIXG6#7%7$#!\"(\"\"'#\"\"#\"\"$7$#!\"\"F.#\"
\"\"F.7$#\"\"&F.#F4F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "F.%  #it is an inverse on the right" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")#4
b`$-%'MATRIXG6#7$7$\"\"\"\"\"!7$F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 84 "%%.F;  #but not on the left.  (notice the use of
 %% to refer back to the \"inverse\")." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RTABLEG6%\")s'GQ$-%'MATRIXG6#7%7%#\"\"&\"\"'#\"\"\"\"\"$#!\"
\"F.7%F/F/F/7%F2F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "Matrix([[1,1],[2,2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA
BLEG6%\")#4UT$-%'MATRIXG6#7$7$\"\"\"F,7$\"\"#F.%'MatrixG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "MatrixInverse(%);  #here we have a \+
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r, (in LinearAlgebra:-LA_Main:-MatrixInverse) singular matrix\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "rref:=ReducedRowEchelonForm;
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{MPLTEXT 1 0 91 "Matrix([[3,-1,1,1],[2,2,2,3],[1,-2,-1,4]]);  #solve a
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20 "6#-%'RTABLEG6%\"(#R%o&-%'MATRIXG6#7%7&\"\"$!\"\"\"\"\"F.7&\"\"#F0F
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8 "rref(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(k0j&-%'M
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2*y+2*z=3,x-2*y-z=4\});  #solve the same system lazily." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<%/%\"zG#!#T\"\"%/%\"yG\"\"'/%\"xG#\"#BF(" }}}
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