MTHG371--Quiz 5--Solutions

Problem 1

A,B,C 4 by 4 matrices
Det[A]=2; Det[B]=-1; Det[C]=0;

(* A invertible: yes *)
(* B invertible: yes *)
(* C invertible: no *)
(* A orthogonal: no *)
(* B orthogonal: maybe *)
(* C orthogonal: no *)

Det[A . B . Transpose[A]] = 2*(-1)*2 = -4

Det[3 . A. A] = 3^4 * 2^2 = 324

Problem 2

A={{1,0,1},{0,1,0},{1,0,1}}

{{1, 0, 1}, {0, 1, 0}, {1, 0, 1}}

Eigenvalues[A]

{2, 1, 0}

S=Transpose[Eigenvectors[A]]

{{1, 0, -1}, {0, 1, 0}, {1, 0, 1}}

d=DiagonalMatrix[Eigenvalues[A]]

{{2, 0, 0}, {0, 1, 0}, {0, 0, 0}}

A==S.d.Inverse[S]

True

Problem 3

A=DiagonalMatrix[{3,-4}];  
    (* For simplicity, I will pretend A is diagonal *)

Det[t*IdentityMatrix[2]-A]

-12 + t + t^2

Tr[A]

-1

Det[A]

-12

Det[4*IdentityMatrix[2]+A]

0

diag = DiagonalMatrix[{3, -4}]

{{3, 0}, {0, -4}}

B = {{1, 3}, {3, -2}} ; {Tr[B], Det[B]}    (* So B not similar to A *)

{-1, -11}

c = {{5, 6}, {-3, -6}} ; {Tr[c], Det[c]}     (* So C is similar to A, since both have same (simple) eigenvalues  *)

{-1, -12}

M = {{3, 2}, {3, -2}} ; {Tr[M], Det[M]}    (* So M not similar to A *)

{1, -12}

Problem 4

S={{5,2},{9,4}};  MatrixForm[S]

( 5 2 ) 9 4

d={{6,0},{0,7}}

{{6, 0}, {0, 7}}

Inverse[S]

{{2, -1}, {-9/2, 5/2}}

A=S.d.Inverse[S]

{{-3, 5}, {-18, 16}}

Created by Mathematica  (April 19, 2005)