Math 234, Practice Exam 3 Key 1.) [10 points] Find the solution to the linear system of equations, if it exists.

10242 321 −=+− xxx

235 31 =+ xx

246 321 =+− xxx

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2.) [10 points] Given the following matrices.

−=

33

02A

−−=

163

014B

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

a.) AB

b.) BA

c.) TBB

d.) 2A

e.) 2B

f.) BA 32 −

g.) AA 1−

h.) |�| i.) |�|

Determinants are only defined for square matrices.

Since we know ���� = �, there is no need to compute ���.

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3.) [10 points] ] Find the inverse of the matrix below, if it exists. Show all work.

−

−

=

230

639

402

A

Use your answer above to solve the system ( )3,2,1=xAv

.

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4.) [5 points] Find the eigenvalues and eigenvectors of

−−=

16

14A

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

−=

61

24A

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6.) [5 points] Find the eigenvalues of the matrix

−

=

300

420

216

A .

Then find the eigenvector corresponding to the smallest eigenvalue.

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7.) [5 points] Solve the initial value problem

�= 2 3−1 2� ��, ���0� = 3−4�

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