Page 149 - Schaum's Outline of Differential Equations

132 MATRICES [CHAP. 15

SCALAR AND MATRIX MULTIPLICATION
If A, is either a number (also called a scalar) or a function of t, then AA (or, equivalently, AA) is defined to
be the matrix obtained by multiplying every element of A by A. That is,

Let A = [fly] and B = [fo y] be two matrices such that A has r rows and n columns and B has n rows and p
columns. Then the product AB is defined to be the matrix C = [c y] given by

The element c y is obtained by multiplying the elements of the rth row of A with the corresponding elements of
thej'th column of B and summing the results.
Matrix multiplication is associative and distributes over addition; in general, however, it is not commutative.
Thus,

A(BC) = (AB)C, A(B + C) = AB + AC, and (B + C)A = BA + CA

but, in general,

POWERS OF A SQUARE MATRIX
If n is a positive integer and A is a square matrix, then

2
3
In particular, A = AA and A = AAA. By definition, A° = I, where

is called an identity matrix. For any square matrix A and identity matrix I of the same size
AI = IA = A

DIFFERENTIATION AND INTEGRATION OF MATRICES

The derivative of A = [a y] is the matrix obtained by differentiating each element of A; that is,

Similarly, the integral of A, either definite or indefinite, is obtained by integrating each element of A. Thus,

and