Page 149 - 35Linear Algebra

7.4 Review Problems 149

Give a few simple examples before you start explaining.

8. Compute exp(A) for the following matrices:

λ 0
• A =
0 λ

1 λ
• A =
0 1

0 λ
• A =
0 0

Hint

 
1 0 0 0 0 0 0 1
0 1 0 0 0 0 1 0
 
 
0 0 1 0 0 1 0 0
 
 
0 0 0 1 1 0 0 0
 
9. Let M =   . Divide M into named blocks,
0 0 0 0 2 1 0 0
 
 
0 0 0 0 0 2 0 0
 
 
0 0 0 0 0 0 3 1
 
0 0 0 0 0 0 0 3
with one block the 4 × 4 identity matrix, and then multiply blocks to
2
compute M .
T
10. A matrix A is called anti-symmetric (or skew-symmetric) if A = −A.
Show that for every n × n matrix M, we can write M = A + S where
A is an anti-symmetric matrix and S is a symmetric matrix.
T
T
Hint: What kind of matrix is M + M ? How about M − M ?
11. An example of an operation which is not associative is the cross prod-
uct.

(a) Give a simple example of three vectors from 3-space u, v, w such
that u × (v × w) 6= (u × v) × w.

149