# 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 Ab_{1} is a linear combination of the columns of A
and

Ab_{2} 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 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)

**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,

Prev | Next |