1. Answer the following questions

A. Is lambda = 2 an eigenvalue of the following matrix? Why or why not?
3 2
3 8

B. Is lambda - -2 an eigenvalue of the following matrix? Why or why not?
7 3
3 -1

C. Is the following vector an eigenvector of the following matrix? If so, find the eigenvalue.
1
4

D. -3 1
-3 8

2. Find a basis for the eigenspace corresponding to each listed eigenvalue and the following matrix

a. lambda = 1, 5
5 0
2 1

b. lambda = 4
10 -9
4 -2

3. Find the eigenvalues of the following matrices

a.
0 0 0
0 2 5
0 0 -1

b.
4 0 0
0 0 0
1 0 -3

4. In the following problem, A is an n x n (square) matrix. Mark each statement True or False. Justify your answer

a. If Ax = lambda x for some vector x, then lambda is an eigenvalue of A.

b. A matrix A is not invertible if and only if 0 is an eigenvalue of A.

c. A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a non-trivial solution

d. Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.

e. To find the eigenvalues of A, reduce A to echelon form.

5. Diagonalize the following matrices if possible

a.
1 0
6 -1

b.
5 1
0 5

c.
2 3
4 1