Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.4 BASIC MATRIX ALGEBRA

except for thejth, which equals unity, then we find

Aej=











A 11 A 12 ... A 1 N

A 21 A 22 ... A 2 N

..

.

..

.

..

.

..

.

AM 1 AM 2 ... AMN

































0

0

..

.

1

..

.

0























=











A 1 j

A 2 j

..

.

AMj











,

and so confirm our identification of the matrix elementAijas theith component

ofAejin this basis.

From (8.28) we can extend our discussion to the product of two matrices

P=AB,wherePis the matrix of the quantities formed by the operation of

the rows ofAon the columns ofB, treating each column ofBin turn as the

vectorxrepresented in component form in (8.32). It is clear that, for this to be

a meaningful definition, the number of columns inAmust equal the number of

rows inB. Thus the productABof anM×NmatrixAwith anN×RmatrixB

is itself anM×RmatrixP,where

Pij=

∑N

k=1

AikBkj fori=1, 2 ,...,M, j=1, 2 ,...,R.

For example,P=ABmay be written in matrix form

(

P 11 P 12

P 21 P 22

)

=

(

A 11 A 12 A 13

A 21 A 22 A 23

)





B 11 B 12

B 21 B 22

B 31 B 32





where

P 11 =A 11 B 11 +A 12 B 21 +A 13 B 31 ,

P 21 =A 21 B 11 +A 22 B 21 +A 23 B 31 ,

P 12 =A 11 B 12 +A 12 B 22 +A 13 B 32 ,

P 22 =A 21 B 12 +A 22 B 22 +A 23 B 32.

Multiplication of more than two matrices follows naturally and is associative.

So, for example,

A(BC)≡(AB)C, (8.34)

provided, of course, that all the products are defined.

As mentioned above, ifAis anM×Nmatrix andBis anN×Mmatrix then

two product matrices are possible, i.e.

P=AB and Q=BA.