1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 339

For any matrix A , a A = A a. Hence, the following two distributive laws

are valid for scalar multiplication

a(A + B) = a A + a B = A a +Ba= (A+ B)a

(a+ (3)A = a A + (3A = Aa + A(3 = A(a + (3).

DEFINITION Matrix Multiplication

When the number of columns of the matrix A is equal to the number of

rows of the matrix B , the matrix product AB is defined. If A is an m x n

matrix and B is an n x p matrix, then the matrix product AB = C = ( Cij)

is an m x p matrix with entries

n

(1) Cij = L aikbkj = ail bij + ai2b2j + · · · + ainbnj.

k=l

For example, let

and

Observe that A is a 2 x 2 matrix and B is a 2 x 3 matrix, so the matrix
product AB will be a 2 x 3 matrix. Using equation (1) to compute each entry
of the product, we find

(

1(-3) + 2(2)

c = -3( -3) + 5(2)

1(1) + 2(0)

-3(1) + 5(0)

1(2) + 2(-3)) ( 1 1 -4)

-3(2) + 5(-3) - 19 -3 -21.

Notice, in this example, that the matrix product BA is undefined, since the
number of columns of B - namely, 3- is not equal to the number of rows of
A - which is 2. Hence, this example illustrates the important fact that matrix
multiplication is not commutative. That is, in general

AB-/:-BA.

In order for both products AB and BA to exist and be the same size, it
is necessary that A and B both be square matrices of the same size. Even
then the matrix product AB may not equal the matrix product BA as the