- Which of the following must be true if
*A*, a 2x2 matrix, has a nonzero eigenvector **a** satisfying
*A***a**= 5**a**?

a) *A*'s eigenvectors are all of R^{2}

b) *A* must have at least an entire line through the origin in R^{2} as it's eigenvectors, where the vectors get stretched by 5.

c) *A* can have just the one eigenvector **a** that is stretched by 5.

d) *A* has two eigenvectors, the second being from Maple

e) none of the above

- If the reduced augmented matrix for the system (A-lambdaI)x=0 is

Matrix([[0,0,0],[0,0,0]]),

with
*A* as a 2x2 matrix

then the non-zero (real) eigenvectors of A are:

a) none

b) a line through the origin

c) all of R^{2}

d) a subspace of R^{3} (with 3 coordinates)

e) none of the above

- What are the non-zero real eigenvectors of

Matrix([[cos(Pi),-sin(Pi)],[sin(Pi),cos(Pi)]])?

a) none

b) a line of eigenvectors

c) two different lines of eigenvectors

d) all of R^{2}

e) none of the above

- If we write a basis for the eigenspace of

Matrix([[cos(Pi),-sin(Pi)],[sin(Pi),cos(Pi)]]), how many vectors does
it have?

a) 0

b) 1

c) 2

d) infinite

e) none of the above

- What are the non-zero real eigenvectors of

Matrix([[cos(Pi/5),-sin(Pi/5)],[sin(Pi/5),cos(Pi/5)]])?

a) none

b) a line of eigenvectors

c) two different lines of eigenvectors

d) all of R^{2}

e) none of the above

Solutions

1. b)

2. c)

3. d)

4. c)

5. a)