Page 360 - 35Linear Algebra

360 Sample Final Exam

 
−1
so  0  is an eigenvector.
1
For λ = −3

     
2 −1 0 1 −1 1 1 0 −1
M − (−3).I =  −1 1 −1  ∼  0 1 −2  ∼  0 1 −2  ,
0 −1 2 0 −1 2 0 0 0

 
1
so   is an eigenvector.
2
1
√
(c) The characteristic frequencies are 0, 1, 3.

(d) The orthogonal change of basis matrix
 1 1 1 
√ − √ √
3 2 6
− √ 0
 1 √ 2 
3 6
P =  
√ 1 √ 1 √ 1
3 2 6
It obeys MP = PD where
 
0 0 0
D =  0 −1 0  .
0 0 −3

 
1
(e) Yes, the direction given by the eigenvector  −1  because its eigen-
1
value is zero. This is probably a bad design for a bridge because it can
be displaced in this direction with no force!

a b
2
T
2
12. (a) If we call M = , then X MX = ax + 2bxy + dy . Similarly
b d

c
T
T
putting C = yields X C + C X = 2X C = 2cx + 2ey. Thus
e
2
2
0 = ax + 2bxy + dy + 2cx + 2ey + f

a b x c x

= x y + x y + c e + f .
b d y e y
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