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
1
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 ���.
2
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
.
3
4.) [5 points] Find the eigenvalues and eigenvectors of
−−=
16
14A
4
5.) [5 points] Find the eigenvalues and eigenvectors of the matrix below.
−=
61
24A
5
6.) [5 points] Find the eigenvalues of the matrix
−
=
300
420
216
A .
Then find the eigenvector corresponding to the smallest eigenvalue.
6
7.) [5 points] Solve the initial value problem
�= 2 3−1 2� ��, ���0� = 3−4�
7