1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations

The essential algebraic properties for matrices are stated below.

DEFINITION Equality of Matrices

Two matrices A = ( aij) and B = ( bij) are equal if and only if

(i) they are the same size

and

(ii) % = bij for all i and j.

DEFINITIONS Addition and Subtraction of Matrices

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In order to add or subtract two matrices they must necessarily be the

same size.

If A= (aiJ) and B = (biJ) are both m x n matrices, then

(i) the sum S = A+ B is an m x n matrix with elements Sij = aiJ + bij-

that is,

and

(ii) the difference D = A - B is an m x n matrix with elements dij =

aiJ - bij- that is,

For example, if

A=

G

-:) and B~H

-D,

then

(2+3 -1+4) ( 5

A+B= O+V2° i-2 = V2 i~2)

7r + 4 e+O n+4

and

A- B= c-3 O-V2° -i + 2 1-4) = (-1 -V2 i:2 -5).

7r - 4 e-0 n-4