1550078481-Ordinary_Differential_Equations__Roberts_

336 Ordinary Differential Equations

DEFINITION Size of a Matrix

If the matrix A h as m rows and n columns, we will write A is an m x n

matrix- where m x n is read "m by n." We also say A has size m x n.

For our purposes the elements or entries of a matrix will be real numbers,

complex numbers, or functions. An element of an m x n matrix A which is

in the ith row and jth column is denoted by aij· Hence, the matrix A may

b e represented in any one of the following three equivalent ways.

A=

The 2 x 3 matrix

(

-1

B = -5i

is called a constant matrix because each entry is a constant.

A square matrix is a matrix wit h the same number of rows as columns

( m = n). The square, 2 x 2 matrix

c = ( 1

2x

3ex)

sinx

in which each element is a function of x is often written as C(x) to indicate
that the entries of t he matrix a re functions. A column vector is am x 1
matrix and a row vector is a 1 x n matrix. We will denote a column vector
wit h a bold-faced, lowercase letter such as a , b , c , .... The 3 x 1 matrix

is an example of a constant column vector. And the 2 x 1 matrix

d(x) = (3x -1)

tanx

is an example of a column vector whose entries are functions.