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Compute the eigenvalues and corresponding eigenvectors of the matrix A

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Cornelis VAN DER MEE, Spring 2008, Math 3330, Sample Exam 4 Name: . . . Grade: . . . Rank: . . . . To receive full credit, show all of your work. Neither calculators nor computers are allowed.

1. Compute the eigenvalues and corresponding eigenvectors of the matrix

A = 3 −2 15 −10

 .

Use this information to diagonalize the matrix A if possible. Otherwise indicate why diagonalization is not possible.

2. Find a 2 × 2 matrix A such that 1 2



and 2 3



are eigenvectors of A, with eigenvalues −2 and 3, respectively.

3. Find the solution of the discrete dynamical system x(n + 1) = Ax(n), n = 0, 1, 2, 3, . . . , where

A =2 3 0 1



, x(0) =0 1

 .

4. Find all eigenvalues (real and complex) of the matrix

A =

0 1 0

0 0 1

4 −4 1

.

Why or why not is it possible to diagonalize the matrix A?

5. Compute the eigenvalues and corresponding eigenvectors of the matrix

A =

0 0 0 1

0 3 −3 1

0 1 0 0

0 0 1 0

 .

Use this information to diagonalize the matrix A if possible. Otherwise indicate why diagonalization is not possible.

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6. Compute the eigenvalues (real and complex) and corresponding eigen- vectors of the matrix

A =

0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0

 .

Explain why your result is in full agreement with the values of Tr(A) and det(A).

7. Why is 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 an eigenvalue corresponding to any 9 × 9 sudoku matrix? What is the corresponding eigenvector?

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