1550078481-Ordinary_Differential_Equations__Roberts_

340 Ordinary Differential Equations

following simple example illustrates. Let

Then

and

A=(~ ~) and B = G ~).

AB= (0

1

1) (1 0) (0(1) + 1(0) 0(0) + 1(0)) (0 0)

0 0 0 - 1(1)+ 0 (0) 1(0)+0(0) - 1 0

BA= (1 0) (0 1) = (1(0) + 0(1) 1(1) + 0(0)) = (0 1)

0 0 1 0 0(0)+0(1) 0(1)+0(0) 0 0.

So, even if A and B are both square matrices of the same size, AB need not
equal BA.

Although matrix multiplication is not commutative, it is associative and
both left and right distributive. That is, if A is an n x m matrix, B is an
m x p matrix, and C is a p x q matrix, t hen

A (BC) = (AB)C.

If A is an n x m matrix, B is an m x p matrix, C is an m x p matrix, and D
is a p x q matrix, then

A(B + C) = AB +AC and (B + C)D = BD + BD.

DEFINITION Identity Matrix

The n x n identity matrix, I , is the matrix

11 0 0 0

0 1 0 0

I= 0 0 1 0

\0 0 0 1

If A is any n x n square matrix and I is the n x n identity, then by the
definition of matrix multiplication

AI=IA=A.