Math 234, Spring 2014 Exam 3 Key 1.) [9 points] Given the following matrices.

−=

14

01

23

A

−−=

12

30B

Compute the follwing quantities, if they exist. If the quantity is undefined, explain why.

a.) BA+

b.) AB

c.) BA

1

2.) [11 points] Find the solution to the linear system, if it exists. Show all work.

42 31 =− xx

44 321 =++ xxx

93 32 =− xx

2

3.) [8 points] Find the determinant of the matrix below.

−

−

−

=

230

024

131

A

3

4.) [5 points] Find the inverse of the matrix below, if it exists.

−

−=

43

12A

4

5.) [6 points] Find the eigenvalues and eigenvectors of

=

33

28A

5

6.) [5 points] Find the general solution of the DE system

��� = �3 19 −5� ��

6

7.) [6 points] Consider the ODE system ��� = ��� where A is a 2x2 matrix. Listed below are

several possible eigenvalues of A. Match each pair of eigenvalues with the most reasonable

phase portrait by writing the letter in the blank. Each letter is used only once.

______ 2 ,3=λ

______ 2 ,3 −=λ

______ 2 ,3 −−=λ

______ i23±=λ

______ i23±−=λ

______ i2±=λ

A

B

C

D

E

F