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t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7hq%$AddG%(AdjointG%3BackwardSubst
ituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-BilinearF
ormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0Colum
nDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0Conditi
onNumberG%/ConstantMatrixG%/ConstantVectorG%2CreatePermutationG%-Cross
ProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/DiagonalMatrixG%*Di
mensionG%+DimensionsG%+DotProductG%,EigenvaluesG%-EigenvectorsG%&Equal
G%2ForwardSubstituteG%.FrobeniusFormG%4GaussianEliminationG%2GenerateE
quationsG%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5Given
sRotationMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3Hermitian
TransposeG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/Ident
ityMatrixG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG
%*IsUnitaryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositi
onG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%.MatrixInver
seG%5MatrixMatrixMultiplyG%+MatrixNormG%5MatrixScalarMultiplyG%5Matrix
VectorMultiplyG%2MinimalPolynomialG%&MinorG%)MultiplyG%,NoUserValueG%%
NormG%*NormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG
%0QRDecompositionG%-RandomMatrixG%-RandomVectorG%%RankG%6ReducedRowEch
elonFormG%$RowG%-RowDimensionG%-RowOperationG%)RowSpaceG%-ScalarMatrix
G%/ScalarMultiplyG%-ScalarVectorG%*SchurFormG%/SingularValuesG%*SmithF
ormG%*SubMatrixG%*SubVectorG%)SumBasisG%0SylvesterMatrixG%/ToeplitzMat
rixG%&TraceG%*TransposeG%0TridiagonalFormG%+UnitVectorG%2VandermondeMa
trixG%*VectorAddG%,VectorAngleG%5VectorMatrixMultiplyG%+VectorNormG%5V
ectorScalarMultiplyG%+ZeroMatrixG%+ZeroVectorG%$ZipG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "#Problem 1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "Matrix(3,1,[[3],[4],[5]]);  #column vector as a 3 by \+
1 matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")k\"eh\"-%'M
ATRIXG6#7%7#\"\"$7#\"\"%7#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 120 "Matrix(3,1,[[3],[4],[5]]) + Matrix(2,1,[[-1],[2]]);  # you ca
n't add these -- they aren't of the same length. problem 1a" }}{PARA 
8 "" 1 "" {TEXT -1 41 "Error, (in rtable/Sum) invalid arguments\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Matri
x(3,1,[[1],[3],[-2]]) - Matrix(3,1,[[-2],[1],[5]]);  #problem 1b" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"))=Dd\"-%'MATRIXG6#7%7#
\"\"$7#\"\"#7#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=Ma
trix(2,2,[[2,1],[-1,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'
RTABLEG6$\")k*)o:-%'MATRIXG6#7$7$\"\"#\"\"\"7$!\"\"\"\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B:=Matrix(2,3,[[1,-2,0],[-2,2,1]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\")c$)y:-%'MATRI
XG6#7$7%\"\"\"!\"#\"\"!7%F/\"\"#F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "MatrixMatrixMultiply(A,B);  #problem 1b -- it works" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")7>&e\"-%'MATRIXG6#7$7
%\"\"!!\"#\"\"\"7%!\"(\"\")\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "MatrixMatrixMultiply(B,A);  # problem 1c -- it doesn'
t work." }}{PARA 8 "" 1 "" {TEXT -1 117 "Error, (in LinearAlgebra:-Mat
rixMatrixMultiply) first matrix column dimension (3) <> second matrix \+
row dimension (2)\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#Pro
blem 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 86 "A:=Matrix(3,4,[[1,1,1,4],[1,-2,1,1],[2,3,
4,13]]);  #set up the augmented matrix for 2a" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")7!4f\"-%'MATRIXG6#7%7&\"\"\"F.F.\"
\"%7&F.!\"#F.F.7&\"\"#\"\"$F/\"#8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "A:=RowOperation(A,[2,1],-1);  #add -1 times row 1 to \+
row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")Go'e\"-%
'MATRIXG6#7%7&\"\"\"F.F.\"\"%7&\"\"!!\"$F1F27&\"\"#\"\"$F/\"#8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:=RowOperation(A,[3,1],-2);
 #add -2 times row 1 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A
G-%'RTABLEG6$\")SC%f\"-%'MATRIXG6#7%7&\"\"\"F.F.\"\"%7&\"\"!!\"$F1F27&
F1F.\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:=RowOpe
ration(A,[2,3]); #swap rows 2 and 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"AG-%'RTABLEG6$\");Q,;-%'MATRIXG6#7%7&\"\"\"F.F.\"\"%7&\"\"!F.\"\"
#\"\"&7&F1!\"$F1F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "A:=Ro
wOperation(A,[3,2],3);  #add 3 times row 2 to row 3" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")g73;-%'MATRIXG6#7%7&\"\"\"F.F.\"
\"%7&\"\"!F.\"\"#\"\"&7&F1F1\"\"'\"#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "z:=solve(6*z=12,z);  #set z equal to the solution of \+
6z=12" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 102 "y:=solve(y+2*z=5,y);  #set y equal to th
e solution of y+2z=5 (Maple remembers the value assigned to z)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "x:=solve(x+y+z=4,x);  #solve for x" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"xG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "Matrix(3,1,[[x],[y],[z]]);  #display the solution in \+
vector (column matrix) format" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RT
ABLEG6$\")opF;-%'MATRIXG6#7%7#\"\"\"F+7#\"\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 73 "x:=evaln(x); y:=evaln(y); z:=evaln(z);  #forget \+
values of these variables" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"zGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
#problem 2 part b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=Matrix(2,4,[[1,-1,1,4],[2
,-1,-1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")3
(yj\"-%'MATRIXG6#7$7&\"\"\"!\"\"F.\"\"%7&\"\"#F/F/F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "A:=RowOperation(A,[2,1],-2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")WMP;-%'MATRIXG6#7$7&\"\"\"
!\"\"F.\"\"%7&\"\"!F.!\"$!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "y:=solve(y-3*z=-7,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG,
&%\"zG\"\"$\"\"(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x:
=solve(x-y+z=4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&%\"zG\"
\"#\"\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x:=evaln(x
); y:=evaln(y);z:=evaln(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF
$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"zGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "
Matrix(3,1,[[x],[y],[z]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABL
EG6$\")gjR;-%'MATRIXG6#7%7#%\"xG7#%\"yG7#%\"zG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "z*Matrix(3,1,[[2],[3],[1]]) + Matrix(3,1,[[-3]
,[-7],[0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"zG\"\"\"-%'RTAB
LEG6$\")GyT;-%'MATRIXG6#7%7#\"\"#7#\"\"$7#F&F&F&-F(6$\"))GAk\"-F,6#7%7
#!\"$7#!\"(7#\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "#yo
u can also write it this way -- Maple won't compute this written this \+
way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "#Problem 3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "#set up
 the matrix to compute the inverse" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=Matr
ix(2,4,[[1,3,1,0],[5,1,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
AG-%'RTABLEG6$\"(#**)e%-%'MATRIXG6#7$7&\"\"\"\"\"$F.\"\"!7&\"\"&F.F0F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=RowOperation(A,[2,1]
,-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"(_#*Q%-%'
MATRIXG6#7$7&\"\"\"\"\"$F.\"\"!7&F0!#9!\"&F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "A:=RowOperation(A,2,-1/14);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"()=%p%-%'MATRIXG6#7$7&\"\"\"\"\"$F.
\"\"!7&F0F.#\"\"&\"#9#!\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "A:=RowOperation(A,[1,2],-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'RTABLEG6$\");W17-%'MATRIXG6#7$7&\"\"\"\"\"!#!\"\"\"#9#\"\"$F2
7&F/F.#\"\"&F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#Use Ma
ple to compute the inverse directly " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B:=Matr
ix(2,2,[[1,3],[5,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTA
BLEG6$\")s<s7-%'MATRIXG6#7$7$\"\"\"\"\"$7$\"\"&F." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 58 "C:=MatrixInverse(B);  #notice that we have t
he same result" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6$\"
)KAP9-%'MATRIXG6#7$7$#!\"\"\"#9#\"\"$F07$#\"\"&F0F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "#Finally, use the matrix inverse to solve
 the given system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "MatrixMatrixMultiply(C,Matri
x(2,1,[[6],[16]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")
#HPg\"-%'MATRIXG6#7$7#\"\"$7#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "solve(\{x+3*y=6,5*x+y=16\});  #check" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<$/%\"yG\"\"\"/%\"xG\"\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "#Problem 4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=Matr
ix(3,1,[[1],[-1],[3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RT
ABLEG6$\")?y(H\"-%'MATRIXG6#7%7#\"\"\"7#!\"\"7#\"\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "B:=Matrix(3,1,[[-2],[2],[4]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\")sfG8-%'MATRIXG6#7%7#!\"
#7#\"\"#7#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "C:=Matri
x(3,1,[[8],[-8],[-6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RT
ABLEG6$\")g(3L\"-%'MATRIXG6#7%7#\"\")7#!\")7#!\"'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "#set up the matrix to solve the problem" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "E:=Matrix(3,4,[[1,-2,8,0],[-1,2,-8,0],[3,4,-6,0]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6$\")'4CN\"-%'MATR
IXG6#7%7&\"\"\"!\"#\"\")\"\"!7&!\"\"\"\"#!\")F17&\"\"$\"\"%!\"'F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "E:=RowOperation(E,[2,1],1);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6$\")oB*Q\"-%'MATR
IXG6#7%7&\"\"\"!\"#\"\")\"\"!7&F1F1F1F17&\"\"$\"\"%!\"'F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "E:=RowOperation(E,[2,3]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6$\")w7[9-%'MATRIXG6#7%7&\"
\"\"!\"#\"\")\"\"!7&\"\"$\"\"%!\"'F17&F1F1F1F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "E:=RowOperation(E,[2,1],-3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6$\")#RGZ\"-%'MATRIXG6#7%7&\"\"\"!\"
#\"\")\"\"!7&F1\"#5!#IF17&F1F1F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "z:=1;  #choose an arbitrary value for z" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"zG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "y:=solve(10*y-30*z=0,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "x:=solve(x-2*y+8*z=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG
!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x:=evaln(x); y:=eva
ln(y); z:=evaln(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"zGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "-2*A+3*B+
1*C;  #it works..." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(!
)y.&-%'MATRIXG6#7%7#\"\"!F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "#Problem 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "det([[1,-2,3],[-1,-1,2],[3,0
,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 59 "# since the determinant is 0, the null space \+
is nontrivial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A:=Matrix(3,4,[[1,-2,3,0],[-
1,-1,2,0],[3,0,-1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RT
ABLEG6$\")_p%H\"-%'MATRIXG6#7%7&\"\"\"!\"#\"\"$\"\"!7&!\"\"F3\"\"#F17&
F0F1F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=RowOperation
(A,[2,1],1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")W
]F8-%'MATRIXG6#7%7&\"\"\"!\"#\"\"$\"\"!7&F1!\"$\"\"&F17&F0F1!\"\"F1" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=RowOperation(A,[3,1],-3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")g'fN\"-%'MA
TRIXG6#7%7&\"\"\"!\"#\"\"$\"\"!7&F1!\"$\"\"&F17&F1\"\"'!#5F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=RowOperation(A,[3,2],2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\");$oQ\"-%'MATR
IXG6#7%7&\"\"\"!\"#\"\"$\"\"!7&F1!\"$\"\"&F17&F1F1F1F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "y:=solve(-3*y+5*z=0,y);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"yG,$%\"zG#\"\"&\"\"$" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "x:=solve(x-2*y+3*z=0,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"xG,$%\"zG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "Matrix(3,1,[[x],[y],[z]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6$\")?p]9-%'MATRIXG6#7%7#,$%\"zG#\"\"\"\"\"$7
#,$F-#\"\"&F07#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "#the n
ullspace is one dimensional and has basis consisting" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "#just of (1/3,5/3,1)^T" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "#Problem 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "u:=D(y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"uG,$-%\"DG6#%\"zG#\"\"&\"\"$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "v:=D(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"vG,$--%#@@G6$%\"DG\"\"#6#%\"zG#\"\"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 41 "D(v)(t)=t*v(t)-2*u(t)+sin(t)*y(t)+cos(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$---%#@@G6$%\"DG\"\"$6#%\"zG6#%\"tG#
\"\"&F+,**&F/\"\"\"---F(6$F*\"\"#F,F.F4F0*&#\"#5F+F4--F*F,F.F4!\"\"*(F
0F4-%$sinGF.F4-F-F.F4F4-%$cosGF.F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 33 "The system of equations wanted is" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "x' = u" }}{PARA 0 "" 0 "" {TEXT -1 
6 "u' = v" }}{PARA 0 "" 0 "" {TEXT -1 27 "v' = tv -2u +sin(t)y+cos(t)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#Problem 7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 15 "The equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 26 "A' =  .75 -(7/30)A+(2/20)B" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "B' =  (7/30)A - (7/20)B" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Initial c
onditions are A(0) = 1.5, B(0) = 1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "#we can solve this" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "A:=evaln(A); B:=evaln(B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGF$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dsolve(\{D(A)(t) = .75 - (
7/30)*A(t) + (2/20)*B(t),  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "D(B)
(t) = (7/30)*A(t)-(7/20)*B(t),A(0)=3/2,B(0)=1\});" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#<$/-%\"AG6#%\"tG,,*&-%$expG6#,$*&,&!#N\"\"\"*$-%%sqrtG6
#\"$&QF2F2F2F(F2#F2\"$?\"F2,&F3#!\"\"\"#6F2F<F2#F2\"\"%**#F2\"#GF2F+F2
F:F2F4F2F2*(F>F2-F,6#,$*&,&\"#NF2F3F2F2F(F2#F<F9F2,&F3#F2F=F2F<F2F2*&#
F2FBF2*(FDF2FKF2F4F2F2F<#\"\"*\"\"#F2/-%\"BGF',(F*F2*&FDF2FKF2F2\"\"$F
2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf(%);  #put it int
o floating point format" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/-%\"BG6#
%\"tG,(-%$expG6#,$F($!+%f[:G\"!#5$!+q^w$y#!\"**&$\"*q^w$yF3\"\"\"-F+6#
,$F($!+RZy^XF0F7F7$\"\"$\"\"!F7/-%\"AGF',(F*$!+!=1nk#F3$\"+++++XF3F7*&
$\"+#>QH`$F0F7F8F7!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "
#notice that the exponential terms all go to zero, and" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "#the limiting quantities of salt in
 tanks A and B are just" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "
#fifteen percent of their volume, as one might expect." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "#Problem 8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "x:=evaln(x); y:=evaln(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"x
GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "equ1:= D(x)(t)=4*x(t)-3*y(t);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%equ1G/--%\"DG6#%\"xG6#%\"tG,&-F*F+\"\"%*&\"\"$
\"\"\"-%\"yGF+F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq
u2:=D(y)(t)=2*x(t)-y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%equ2G/-
-%\"DG6#%\"yG6#%\"tG,&-%\"xGF+\"\"#-F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "x:= t->exp(2*t); y:= t->(2/3)*exp(2*t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrow
GF(-%$expG6#,$9$\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf
*6#%\"tG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$9$\"\"##F2\"\"$F(F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "equ1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$-%$expG6#,$%\"tG\"\"#F*F$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 5 "equ2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$expG
6#,$%\"tG\"\"##\"\"%\"\"$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "# it checks for the first set of solutions." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "
x:=t->exp(t); y:=t->exp(t);  #output here is a little weird" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"yG%$expG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "equ1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#%\"tGF$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "equ2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$expG6#%\"tGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "
#and it also checks for the second set of solutions" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT 
-1 104 "Warning, the previous binding of the name GramSchmidt has been
 removed and it now has an assigned value\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco
mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(
augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch
arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co
ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro
wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv
ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg
ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG
%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG
%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%
'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG
%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'
mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG
%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%
'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith
G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG
%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%
(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "W:= t->det([[exp(2*t),(2/3)*exp(2*t)],[exp(t),exp(t)]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WGf*6#%\"tG6\"6$%)operatorG%&arrow
GF(-%$detG6#7$7$-%$expG6#,$9$\"\"#,$F1#F6\"\"$7$-F26#F5F;F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "W(0);  #since the Wronskian \+
is nonzero at 0, the functions are linearly independent." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "c1:=evaln(c1); c2:=evaln(c2);x:=t->c1*exp(2*t)+c2*exp
(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1GF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#c2GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%
\"tG6\"6$%)operatorG%&arrowGF(,&*&%#c1G\"\"\"-%$expG6#,$9$\"\"#F/F/*&%
#c2GF/-F16#F4F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
y:=t->c1*(2/3)*exp(2*t)+c2*exp(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"yGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%#c1G\"\"\"-%$expG6#,$9$
\"\"#F/#F5\"\"$*&%#c2GF/-F16#F4F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "solve(\{x(0)=1,y(0)=2\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%#c1G!\"$/%#c2G\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "#that might seem too much like magic." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "x(0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#c1G\"
\"\"%#c2GF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y(0)=2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#c1G#\"\"#\"\"$%#c2G\"\"\"F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "#now solve these linear equa
tions for c1 and c2 to get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "#the answer given above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "c1:=-3; c2:=4; mat
rix(2,1,[[x(t)],[y(t)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G!\"
$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7#,&-%$expG6#,$%\"tG\"\"#!\"$*&\"\"%\"\"
\"-F*6#F-F2F27#,&F)!\"#*&F1F2F3F2F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "#the final answer in vector form." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "equ1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$expG6#,$%\"tG\"\"#!
\"'*&\"\"%\"\"\"-F&6#F)F.F.F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "equ2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$expG6#,$%\"tG\"\"#
!\"%*&\"\"%\"\"\"-F&6#F)F.F.F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 58 "#and the final answer does satisfy the original equations!" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "#Problem 9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra
);" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, the assigned name GramSchm
idt now has a global binding\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7hq%
$AddG%(AdjointG%3BackwardSubstituteG%+BandMatrixG%&BasisG%-BezoutMatri
xG%/BidiagonalFormG%-BilinearFormG%5CharacteristicMatrixG%9Characteris
ticPolynomialG%'ColumnG%0ColumnDimensionG%0ColumnOperationG%,ColumnSpa
ceG%0CompanionMatrixG%0ConditionNumberG%/ConstantMatrixG%/ConstantVect
orG%2CreatePermutationG%-CrossProductG%-DeleteColumnG%*DeleteRowG%,Det
erminantG%/DiagonalMatrixG%*DimensionG%+DimensionsG%+DotProductG%,Eige
nvaluesG%-EigenvectorsG%&EqualG%2ForwardSubstituteG%.FrobeniusFormG%4G
aussianEliminationG%2GenerateEquationsG%/GenerateMatrixG%2GetResultDat
aTypeG%/GetResultShapeG%5GivensRotationMatrixG%,GramSchmidtG%-HankelMa
trixG%,HermiteFormG%3HermitianTransposeG%/HessenbergFormG%.HilbertMatr
ixG%2HouseholderMatrixG%/IdentityMatrixG%2IntersectionBasisG%+IsDefini
teG%-IsOrthogonalG%*IsSimilarG%*IsUnitaryG%2JordanBlockMatrixG%+Jordan
FormG%(LA_MainG%0LUDecompositionG%-LeastSquaresG%,LinearSolveG%$MapG%%
Map2G%*MatrixAddG%.MatrixInverseG%5MatrixMatrixMultiplyG%+MatrixNormG%
5MatrixScalarMultiplyG%5MatrixVectorMultiplyG%2MinimalPolynomialG%&Min
orG%)MultiplyG%,NoUserValueG%%NormG%*NormalizeG%*NullSpaceG%3OuterProd
uctMatrixG%*PermanentG%&PivotG%0QRDecompositionG%-RandomMatrixG%-Rando
mVectorG%%RankG%6ReducedRowEchelonFormG%$RowG%-RowDimensionG%-RowOpera
tionG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarVectorG%*Schur
FormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVectorG%)SumBasisG%
0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0TridiagonalFor
mG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAngleG%5VectorM
atrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroMatrixG%+ZeroV
ectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:=evaln(x)
; y:=evaln(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "equ1:=D(x)(t)=5*x(t)+4*y(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%equ1G/--%\"DG6#%\"xG6#%\"tG,&-F*F+\"\"&*&\"\"%\"\"\"
-%\"yGF+F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "equ2:=D(y)(
t)=-6*x(t)-5*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%equ2G/--%\"DG
6#%\"yG6#%\"tG,&-%\"xGF+!\"'*&\"\"&\"\"\"-F*F+F3!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve(\{equ1,equ2\});" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#<$/-%\"xG6#%\"tG,&*&%$_C1G\"\"\"-%$expG6#,$F(!
\"\"F,F,*&%$_C2GF,-F.F'F,F,/-%\"yGF',&F*#!\"$\"\"#F2F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#Perhaps that was a little fast...
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lambda:=evaln(lambda);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaGF$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 31 "A:=Matrix(2,2,[[5,4],[-6,-5]]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\")?q?8-%'MATRIXG6#7$7$\"\"&\"
\"%7$!\"'!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A_lambda:=
Matrix(2,2,[[5-lambda,4],[-6,-5-lambda]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)A_lambdaG-%'RTABLEG6$\"))=bK\"-%'MATRIXG6#7$7$,&\"\"
&\"\"\"%'lambdaG!\"\"\"\"%7$!\"',&!\"&F0F1F2" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "det(A_lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,&!\"\"\"\"\"*$)%'lambdaG\"\"#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "solve(det(A_lambda)=0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "lambda
:=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "MatrixMatrixMultiply(A_lambda,Matr
ix(2,1,[[a],[b]]));  #this vector will be zero iff [[a],[b]] is an eig
envector for this value of lambda." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")?VE:-%'MATRIXG6#7$7#,&%\"aG\"\"'*&\"\"%\"\"\"%\"bGF1F17
#,&F-!\"'*&F0F1F2F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"solve(\{a=1,6*a+4*b = 0\});  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%
\"aG\"\"\"/%\"bG#!\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "#so the eigenvector for lambda = -1 is [[1],[-3/2]]." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda:=1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'lambdaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 138 "MatrixMatrixMultiply(A_lambda,Matrix(2,1,[[a],[b]]));  #this \+
vector will be zero iff [[a],[b]] is an eigenvector for this value of \+
lambda." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")7MI;-%'MATRI
XG6#7$7#,&%\"aG\"\"%*&F.\"\"\"%\"bGF0F07#,&F-!\"'*&\"\"'F0F1F0!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{a=1,4*a+4*b = 0\})
;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"bG!\"\"/%\"aG\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "#so the eigenvector for lamb
da = 1 is [[1],[-1]]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "c1*exp(t)*Matrix(2,1,[[1],
[-1]])+c2*exp(-t)*Matrix(2,1,[[1],[-3/2]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&-%$expG6#%\"tG\"\"\"-%'RTABLEG6$\")+*eM\"-%'MATRIXG
6#7$7#!\"$7#\"\"$F)F)*&-F&6#,$F(!\"\"F)-F+6$\")GwI9-F/6#7$7#\"\"%7#!\"
'F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "c1:=evaln(c1); c2:
=evaln(c2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1GF$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#c2GF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "x:=t->c1*exp(t)+c2*exp(-t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%#c1G\"\"\"-%$expG6#9$F/
F/*&%#c2GF/-F16#,$F3!\"\"F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "y:=t->-c1*exp(t)-(3/2)*c2*exp(-t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"yGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%#c1G\"
\"\"-%$expG6#9$F/!\"\"*&#\"\"$\"\"#F/*&%#c2GF/-F16#,$F3F4F/F/F4F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "equ1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,&*&%#c1G\"\"\"-%$expG6#%\"tGF'F'*&%#c2GF'-F)6#,$F+!
\"\"F'F1F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "equ2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%#c1G\"\"\"-%$expG6#%\"tGF'!\"\"*
(#\"\"$\"\"#F'%#c2GF'-F)6#,$F+F,F'F'F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "#so the purported general solution does satisfy the o
riginal equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x:=eva
ln(x); y:=evaln(y);c1:=evaln(c1); c2:=evaln(c2);lambda:=evaln(lambda);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1GF$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2GF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'lambdaGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "equ1:=D(x)(t)=x(t)-y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%equ
1G/--%\"DG6#%\"xG6#%\"tG,&-F*F+\"\"\"-%\"yGF+!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "equ2:=D(y)(t)=x(t)+y(t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%equ2G/--%\"DG6#%\"yG6#%\"tG,&-%\"xGF+\"\"\"-F*F+F0
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve(\{equ1,equ2\});
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"yG6#%\"tG,$*&-%$expGF'\"\"
\",&*&%$_C1GF--%$cosGF'F-F-*&%$_C2GF--%$sinGF'F-!\"\"F-F7/-%\"xGF'*&F+
F-,&*&F0F-F5F-F-*&F4F-F1F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "#Perhaps that was a little fast..." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=Ma
trix(2,2,[[1,-1],[1,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'
RTABLEG6$\")[m'[\"-%'MATRIXG6#7$7$\"\"\"!\"\"7$F.F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "lambda:=evaln(lambda);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'lambdaGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "A_lambda:=Matrix(2,2,[[1-lambda,-1],[1,1-lambda]]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%)A_lambdaG-%'RTABLEG6$\")g8&\\\"-%'MATRIXG6#7
$7$,&\"\"\"F/%'lambdaG!\"\"F17$F/F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "det(A_lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(
\"\"#\"\"\"*&F$F%%'lambdaGF%!\"\"*$)F'F$F%F%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "solve(det(A_lambda)=0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$^$\"\"\"F$^$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "lambda:=1+I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'la
mbdaG^$\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "MatrixM
atrixMultiply(A_lambda,Matrix(2,1,[[a],[b]]));  #this vector will be z
ero iff [[a],[b]] is an eigenvector for this value of lambda." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")omD;-%'MATRIXG6#7$7#,&*
&^#!\"\"\"\"\"%\"aGF0F0%\"bGF/7#,&F1F0*&F.F0F2F0F0" }}}{EXCHG {PARA 0 
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" 1 "" {XPPMATH 20 "6#<$/%\"bG^#!\"\"/%\"aG\"\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "#so the eigenvector for lambda = 1+I is [[1]
,[-I]]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#the eigenvector
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\"-%'MATRIXG6#7$7#,&*&%#c2G\"\"\"-%$expG6#*&^$F1!\"\"F1%\"tGF1F1F1*&%#
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SolG-%'RTABLEG6$\")38r:-%'MATRIXG
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\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G#\"\"\"\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Sol1G-%'RTABLEG6$\")38r:-%'MATRIXG6
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c1:=1/(2*I); c2:=-1/(2*I); Sol2:=So
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"" 1 "" {XPPMATH 20 "6#>%#c2G^##\"\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%Sol2G-%'RTABLEG6$\")38r:-%'MATRIXG6#7$7#*&-%$expG6#%
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0 "" {TEXT -1 97 "I found these real solutions by direct computation f
rom the complex solutions; to use the formula" }}{PARA 0 "" 0 "" 
{TEXT -1 91 "in the book, lambda = 1+I so alpha = 1 and beta =1, and t
he eigenvector [[1][-I]] breaks up" }}{PARA 0 "" 0 "" {TEXT -1 99 "int
o [[1],[0]]+[[0],[-1]]I  so v1 is (1,0)^T and v2 is (0,-1)^T (switchin
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