1550078481-Ordinary_Differential_Equations__Roberts_

338 Ordinary Differential Equations

From the definition of addition, it is evident t hat matrix addition is com-
mutative and associative, since addition of real numbers, complex numbers,
and functions is commutative and associative. Thus, if A , B , and C are all
m x n matrices, then

A+B=B+A (matrix addition is commutative)

A + (B + C) = (A + B) + C (matrix addition is associative)

DEFINITION Zero Matrix

The symbol 0 will be used to denote any matrix whose elements a re all

zero.

Thus, the 2 x 3 zero matrix is

0 = (0 0 0)

0 0 0

and t he 2 x 1 zero matrix (a zero column vector) is

0 = (~).

If A is any m x n matrix and 0 is the m x n zero matrix, then

A+ O = O+ A =A.

DEFINITION Scalar Multiplication

If a is a real number, a complex number, or a scalar function a nd if A is

an m x n matrix, t hen a A is the m x n matrix with elements aaij- that is,

For example,

3 ( 4) - ( -x - -312) x and

(

ex sinx )

iex ~os x.