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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "First load the \"LinearAl
gebra\" package. It contains many useful commands. The \"LinearAlgebra
\" package is different from the \"linalg\" package." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7[r%$AddG%(AdjointG%3BackwardSubstituteG%+BandMatrixG%&
BasisG%-BezoutMatrixG%/BidiagonalFormG%-BilinearFormG%5CharacteristicM
atrixG%9CharacteristicPolynomialG%'ColumnG%0ColumnDimensionG%0ColumnOp
erationG%,ColumnSpaceG%0CompanionMatrixG%0ConditionNumberG%/ConstantMa
trixG%/ConstantVectorG%2CreatePermutationG%-CrossProductG%-DeleteColum
nG%*DeleteRowG%,DeterminantG%/DiagonalMatrixG%*DimensionG%+DimensionsG
%+DotProductG%6EigenConditionNumbersG%,EigenvaluesG%-EigenvectorsG%&Eq
ualG%2ForwardSubstituteG%.FrobeniusFormG%4GaussianEliminationG%2Genera
teEquationsG%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5Gi
vensRotationMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3Hermit
ianTransposeG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/Id
entityMatrixG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimil
arG%*IsUnitaryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompos
itionG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%.MatrixIn
verseG%5MatrixMatrixMultiplyG%+MatrixNormG%5MatrixScalarMultiplyG%5Mat
rixVectorMultiplyG%2MinimalPolynomialG%&MinorG%(ModularG%)MultiplyG%,N
oUserValueG%%NormG%*NormalizeG%*NullSpaceG%3OuterProductMatrixG%*Perma
nentG%&PivotG%*PopovFormG%0QRDecompositionG%-RandomMatrixG%-RandomVect
orG%%RankG%6ReducedRowEchelonFormG%$RowG%-RowDimensionG%-RowOperationG
%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarVectorG%*SchurFormG
%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVectorG%)SumBasisG%0Sylv
esterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0TridiagonalFormG%+U
nitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAngleG%5VectorMatrix
MultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroMatrixG%+ZeroVector
G%$ZipG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Next define the matric
es M1 and M2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "M1 := Matr
ix([[1,0],[0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'RTABLEG
6%\")ggt7-%'MATRIXG6#7$7$\"\"\"\"\"!7$F/F.%'MatrixG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "M2 := Matrix([[3,3],[4,4]]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#M2G-%'RTABLEG6%\")%GVE\"-%'MATRIXG6#7$7$\"\"
$F.7$\"\"%F0%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Next fin
d the sum of M1 and M2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "M
1 + M2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")K^@;-%'MATRI
XG6#7$7$\"\"%\"\"$7$F,\"\"&%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 64 "Next find the product of M1 and M2. Correct syntax must be used
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "M1 * M2;" }}{PARA 8 "" 
1 "" {TEXT -1 45 "Error, (in rtable/Product) invalid arguments\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "M1 . M2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")[Ne8-%'MATRIXG6#7$7$\"\"$F,7$\"\"%F.%'Ma
trixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Next define the row vect
or V1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V1 := Vector[row]
([1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V1G-%'RTABLEG6%\")O?48-
%'VECTORG6#7$\"\"\"\"\"#&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 35 "Next find the product of V1 and M2." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "V1 . M2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'RTABLEG6%\")!Gti\"-%'VECTORG6#7$\"#6F+&%'VectorG6#%$rowG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "Next find the product of M2 and V1." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "M2.V1;" }}{PARA 8 "" 1 "" 
{TEXT -1 271 "Error, (in LinearAlgebra:-MatrixVectorMultiply) invalid \+
input: LinearAlgebra:-MatrixVectorMultiply expects its 2nd argument, v
, to be of type Vector[column] but received Vector[row](2, [...], data
type = anything, storage = rectangular, order = Fortran_order, shape =
 [])\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "What happened in the co
mputation of the product of M2 and V1?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 33 "Next define the column vector V2." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "V2 := Vector[column]([1,2]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#V2G-%'RTABLEG6%\")O1B;-%'MATRIXG6#7$7#\"\"\"7#\"
\"#&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Next \+
find the product of M2 and V2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "M2.V2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")chK;-%
'MATRIXG6#7$7#\"\"*7#\"#7&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 93 "Next find the product of  V1 and M2 and V2. The answ
er is supposed to be a one by one matrix." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "V1.M2.V2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#L" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Define the constant k." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Fin
d the scalar product of k with M1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "k.M1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%
\")C+S;-%'MATRIXG6#7$7$\"\"#\"\"!7$F-F,%'MatrixG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 32 "Find M1 raised to the kth power." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "M2^k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'RTABLEG6%\")7/+7-%'MATRIXG6#7$7$\"#@F,7$\"#GF.%'MatrixG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Find the inverse of M1." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "MatrixInverse(M1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\")+nW;-%'MATRIXG6#7$7$\"\"\"\"\"!7$F-F,%'M
atrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Find the inverse of M2.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MatrixInverse(M2);" }}
{PARA 8 "" 1 "" {TEXT -1 66 "Error, (in LinearAlgebra:-LA_Main:-Matrix
Inverse) singular matrix\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Wha
t happened in the computation of the inverse of M2?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Next obtain a f
ormula for the inverse of a 2 by 2 matrix." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "A := Matrix([[a,b],[c,d]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")c_\\;-%'MATRIXG6#7$7$%\"aG%\"bG7$%
\"cG%\"dG%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Matri
xInverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")?a^;-%'
MATRIXG6#7$7$*&%\"dG\"\"\",&*&%\"aGF.F-F.F.*&%\"bGF.%\"cGF.!\"\"F5,$*&
F3F.F/F5F57$,$*&F4F.F/F5F5*&F1F.F/F5%'MatrixG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 59 "Is there a nice formula for the inverse of a 3 by 3 ma
trix?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "37 0 0
" 59 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }
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