Matrix Operations

Matrix Notation:

Two ways to denote m ·n matrix A:

In terms of the columns of A:

In terms of the entries of A:

Main diagonal entries:___________________

Zero matrix :

THEOREM 1
Let A, B, and C be matrices of the same size, and let r and s
be scalars. Then

Matrix Multiplication
Multiplying B and x transforms x into the vector Bx. In turn, if we
multiply A and Bx, we transform Bx into . So is the
composition of two mappings .

Define the product AB so that

Suppose A is m ·n and B is n ·p where

and

Then

and

Therefore,

and by defining

we have

EXAMPLE: Compute AB where and

Solution :

Note that Ab₁ is a linear combination of the columns of A and
Ab₂ is a linear combination of the columns of A.

EXAMPLE: If A is 4 ·3 and B is 3 ·2, then what are the sizes of
AB and BA?

Solution:

BA would be

which is __________________.

Row-Column Rule for Computing AB (alternate method)

The definition

is good for theoretical work.

When A and B have small sizes, the following method is more
efficient when working by hand.

If AB is defined, let   denote the entry in the ith row and jth
column of AB. Then

EXAMPLE Compute
AB, if it is defined.

Solution: Since A is 2· 3 and B is 3 ·2, then AB is defined and
AB is _____× _____.

THEOREM 2

Let A be m ·n and let B and C have sizes for which the
indicated sums and products are defined.

	(associative law of multiplication )
	(left - distributive law )
for any scalar r	(right- distributive law )
	(identity for matrix multiplication)

WARNINGS
Properties above are analogous to properties of real numbers .
But NOT ALL real number properties correspond to matrix
properties .

1. It is not the case that AB always equal BA. (see Example 7,
page 114)

2. Even if AB =AC, then B may not equal C. (see Exercise 10,
page 116)

3. It is possible for AB =0 even if A≠ 0 and B ≠0. (see
Exercise 12, page 116)

Powers of A

EXAMPLE:

If A is m ·n, the transpose of A is the n · m matrix, denoted by
A^T, whose columns are formed from the corresponding rows of
A.

EXAMPLE:

EXAMPLE: Let Compute
and

Solution:

THEOREM 3

Let A and B denote matrices whose sizes are appropriate for
the following sums and products.
a. (I.e., the transpose of A^T is A
b.
c. For any scalar r,
d. (I.e. the transpose of a product of matrices
equals the product of their transposes in reverse order . )

EXAMPLE: Prove that =_________.

Solution: By Theorem 3d,