MATH 520 — REVIEW FOR FINAL EXAM

(b) For which a, b, c is B positive definite?
(c) For which a, b, c is C positive definite?
(d) Is A positive definite exactly when B is positive definite? If not, give some choice
of a, b, c where this difference occurs.
(e) Is B positive definite exactly when C is positive definite? If not, give some choice
of a, b, c where this difference occurs.
12. Let
⎡
7 14 10 1
⎤
2
⎡
7 10 6
⎤
1
⎢ 5
A = ⎣
2
10
4
9
4
−3
−2
7 ⎥
⎦ ,
4
⎢ 5
S = ⎣
2
9
4
6
4
1 ⎥
⎦ ,
1
1 2 2 −1 2
1 2 3 1
S −1 ⎡
1 −2 3
⎤
−2
⎡
1 2 0 3
⎤
−4
⎢ −1
= ⎣
1
3
−4
−6
10
4 ⎥
⎦ ,
−7
⎢ 0
B = ⎣
0
0
0
1
0
−2
0
3 ⎥
⎦
0
−2 8 −21 16
0 0 0 0 0
where A = SB. Find a basis for each of the four fundamental subspaces for A. No
arithmetic is required here; it has already been done for you.
13. Let A be a rank 1 n×n matrix with n > 1. We know that A = uv T for certain vectors
u, v in R n .
(a) Show that u is an eigenvector for A. Express the corresponding eigenvalue in terms
of u and v.
(b) Show that A 2 = cA for some constant c.
(c) Show that A is diagonalizable if and only if c = 0.
(d) Assume c = 0. Find simple formulas for A 3 , A 4 , . . . , A k for any k ≥ 1.
(e) Assume c = 0. Find a simple formula for e tA . Show that e tA = α(t)I + β(t)A
where α(t) and β(t) are specific functions of t.
4