Matrix Operations

2.1 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 A(Bx) . So A(Bx) is the composition of two mappings.

Define the product AB so that A(Bx)=(AB)x.

Suppose A is m*n and B is n*p where

Then

and

Therefore,

and by defining

we have A(Bx)=(AB)x.

EXAMPLE: Compute AB where

Solution :

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

Each column of AB is a linear combination of the columns of A
using weights from the corresponding columns of B.

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 __________________.

If A is m*n and B is n*p, then AB is m*p.

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 (AB)_ij 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)
	(right-distributive law)
for any scalar r
	(identity for matrix multiplication)

WARNINGS
Properties above are ana logous 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 AB,

Solution:

THEOREM 3

Let A and B denote matrices whose sizes are appropriate for the fol lowing sums and
products.

a. (I.e., the transpose of A^T is A

c. For any scalar r,

(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,