MATH2270-004
Name:
Midterm 3 Answers
November 20, 2015
Instructions: Do all the problems on both sides of each page. Show all your work and box
your answers. If you get stuck on a problem, skip it and come back to it at the end.
h1 0 0i
1. (a) 0 2 0
003
(b) λ = 1, 2, 3
2. [ 40 14 ]
3. (a) Yes
h1i
(b) 1
7
h 2 i
(c) −2
0
1 h1i 1 h 1 i 1 h 1 i
1
, √ −1
(d) √ 1 , √
3 1
6 −2
2 0
1 4. ~x(t) = c1 e2t [ 11 ] + c2 e−2t −3
5. (a) xk = 1.1k [ 13 ] + .5k [ 21 ]
(b) Origin is a saddle. Draw a picture to show trajectories.
h −1 i
6. (a) 1
1
h0i
(b) 1
0
7.
(c) No. There is no basis of eigenvectors.
2h 6 i
7
−2
3
8. No. There is no basis of eigenvectors since the sum of the dimensions of the eigenspaces of A
is less than 6.
9. [ 21 21 ] Any matrix with linearly dependent columns will do (as long as no entries are 0).
10. Expanding in the first row, we get
(2 − λ) [(4 − λ)(2 − λ) + 1] + 1 [(2 − λ) + 1] − 1 [−1 + (4 − λ)]
1