Prof. Alex Suciu
MTH1230 Linear Algebra
Spring 2001
Final Exam: Solutions
Problem 1
A={{1,6,-7},{2,2,1},{5,4,4}};
MatrixForm[A]
NullSpace[A]
Solve[A.{x,y,z}=={-2,1,3},{x,y,z}]
Problem 2
A={{1,2,3,4,5},{6,7,8,9,10},{11,12,13,14,15},
{16,17,18,19,20}};
RowReduce[A]
NullSpace[A]
Length[NullSpace[Transpose[A]]]
Length[NullSpace[A]]
4-Length[NullSpace[Transpose[A]]]
5-Length[NullSpace[A]]
Problem 3
A={{0,1},{1,1},{2,1},{3,1}}; y={2,1,4,6};
Transpose[A].A
Inverse[Transpose[A].A]
PseudoInverse[A]
{m,b}=Inverse[Transpose[A].A].Transpose[A].y
4*m+b
Problem 4
<<LinearAlgebra`Orthogonalization`
GramSchmidt[{{0,3,4},{0,2,1},{1,2,3}}]
Abs[Det[{{0,3,4},{0,2,1},{1,2,3}}]]
Problem 5
A={{Cos[120*Degree],0,-Sin[120*Degree]},{0,-1,0},
{Sin[120*Degree],0,Cos[120*Degree]}}
Det[A]
Transpose[A] . A == IdentityMatrix[3]
Inverse[A]
A.{-2, 5, 1}
N[%]
Problem 6
A=DiagonalMatrix[{-2,1,3,4}];
Det[t*IdentityMatrix[4]-A]
{Tr[A],Det[A]}
Det[-2*A]
Det[A+2*IdentityMatrix[4]]
Eigenvalues[MatrixPower[A,3]]
Det[A]=!=0
Transpose[A].A==IdentityMatrix[4]
Complement[Union[Eigenvalues[A]],Eigenvalues[A]]=={}
Problem 7
S={{4,1},{2,1}}
MatrixForm[S]
d={{-2,0},{0,5}}
Inverse[S]
A=S.d.Inverse[S]
Problem 8
A={{5,6,0},{7,6,0},{0,0,3}}
Eigenvalues[A]
S=Transpose[Eigenvectors[A]]
d=DiagonalMatrix[Eigenvalues[A]]
A==S.d.Inverse[S]
Problem 9
A={{27,-12},{56,-25}};
MatrixForm[A]
Simplify[MatrixPower[A,t]]
S=Transpose[Eigenvectors[A]]
MatrixForm[S]
d=DiagonalMatrix[Eigenvalues[A]]
S.MatrixPower[d,t].Inverse[S]
Converted by Mathematica
June 7, 2001