Math 410 Third Hour Exam June 25, 1996

Your Name:

There are five questions, each worth 20 points, for a total of 100 points. For full credit you must show all your work.

1. a) Decide whether each of the following vectors is an eigenvector for the matrix B = ( \begin{matrix}2 & −4 \\ 3 & −6 \end{matrix} ). For

each eigenvector, give the eigenvalue.

(i) ( \begin{matrix}1 \\ 1 \end{matrix} )
(ii) ( \begin{matrix}2 \\ 1 \end{matrix} )
(iii) ( \begin{matrix}6 \\ 3 \end{matrix} )
(iv) ( \begin{matrix}1 \\ −1 \end{matrix} )
(v) ( \begin{matrix}2 \\ 3 \end{matrix} )

b) Suppose that A is a 2 × 2 matrix with trace 0. If −3 is an eigenvalue of A, what is the other eigenvalue of A? Explain your answer.

2. a) Suppose that A is a 4 × 4 matrix and A has characteristic polynomial (x + 3)^{2}(x − 5)^{2}. Suppose also

that Rank(A + 3I) = 2, Rank ((A + 3I)^{2}) = 1, Rank(A − 5I) = 1, and Rank ((A − 5I)^{2}) = 1.

Write down a 4 × 4 matrix in Jordan form which is similar to A. Explain how you got your answer.

b) Let B = ( \begin{matrix}3 & 0 & 0 & 0 \\ 0 & −3 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 4 \end{matrix} ). Compute e^{B}.

3. Let A = ( \begin{matrix}1 & 0 & −1 \\ 2 & 0 & 2 \\ −1 & 0 & 1 \end{matrix} ).

a) Find the eigenvalues of A and their multiplicities.

b) For each eigenvalue, find a basis for the corresponding space of eigenvectors.

c) Is A diagonalizable? Why or why not?

4. Let A = ( \begin{matrix}2 & 3 \\ 3 & 2 \end{matrix} ). Find an invertible matrix M and a diagonal matrix B such that A = MBM^{−1}. Show your work.

5. Let C = ( \begin{matrix}2 & 2 \\ 2 & 3 \end{matrix} ). Use the power method to find an approximation for the largest eigenvalue of C and a corresponding eigenvector. Carry out your calculations to 4 significant digits and stop after 3 iterations. (It might be wise, if you have time, to check your answer by finding the eigenvalues in some other way.)