## Multiplication of Matrices

DEFINITION 1.2.8 (Matrix Multiplication / Product)   Let be an matrix and be an matrix. The product is a matrix of order with

That is, if and then

Observe that the product is defined if and only if
THE NUMBER OF COLUMNS OF MATHEND000# THE NUMBER OF ROWS OF MATHEND000#

For example, if and then

 (1.2.1) (1.2.2)

Observe the following:
1. In this example, while is defined, the product is not defined.

However, for square matrices and of the same order, both the product and are defined.

2. The product corresponds to operating on the rows of the matrix (see 1.2.1), and
3. The product also corresponds to operating on the columns of the matrix (see 1.2.2).

DEFINITION 1.2.9   Two square matrices and are said to commute if

Remark 1.2.10
1. Note that if is a square matrix of order then Also, a scalar matrix of order commutes with any square matrix of order .
2. In general, the matrix product is not commutative. For example, consider the following two matrices and . Then check that the matrix product

THEOREM 1.2.11   Suppose that the matrices and are so chosen that the matrix multiplications are defined.
1. Then That is, the matrix multiplication is associative.
2. For any
3. Then That is, multiplication distributes over addition.
4. If is an matrix then
5. For any square matrix of order and we have
• the first row of is times the first row of
• for the row of is times the row of
A similar statement holds for the columns of when is multiplied on the right by

Proof. Part 1. Let and Then

Therefore,

Part 5. For all we have

as whenever Hence, the required result follows.

The reader is required to prove the other parts. height6pt width 6pt depth 0pt

EXERCISE 1.2.12
1. Let and be two matrices. If the matrix addition is defined, then prove that . Also, if the matrix product is defined then prove that .
2. Let and Compute the matrix products and
3. Let be a positive integer. Compute for the following matrices:

Can you guess a formula for and prove it by induction?
4. Find examples for the following statements.
1. Suppose that the matrix product is defined. Then the product need not be defined.
2. Suppose that the matrix products and are defined. Then the matrices and can have different orders.
3. Suppose that the matrices and are square matrices of order Then and may or may not be equal.

A K Lal 2007-09-12