Page 38 - A Course in Linear Algebra with Applications

22 Chapter One: Matrix Algebra

by the rule of addition for matrices.

Example 1.2.12
Let A be an TO X n matrix and B an n x p matrix; write Bi,
B 2, ..., B p for the columns of B. Then, using the partition of
B into columns B = [.B^i^l ••• \B P], we have

AB = (AB 1\AB 2\ ... \AB P).

This follows at once from the row-times-column rule of matrix
multiplication.

Exercises 1.2

1. Define matrices

/l 2 3\ /2 1\ /3 0 4\
A= 0 1 -1 , B= 1 2 , C= 0 1 0 .
\2 1 0/ \1 1/ \2 -1 3/

(a) Compute 3A - 2C.
(b) Verify that (A + C)B = AB + CB.
(c) Compute A 2 and A 3 . (d) Verify that (AB) T =
T T
B A .
m n m+n
2. Establish the laws of exponents: A A = A and
m n
(A ) = A mn where A is any square matrix and TO and n are
non-negative integers. [Use induction on n : see Appendix.]
3. If the matrix products AB and BA both exist, what can
you conclude about the sizes of A and Bl

4. If A = ( 1, what is the first positive power of A

that equals I-p.

5. Show that no positive power of the matrix I J equals

h •