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]