instance, if is 2 x 3 and is 3 x 2, then is defined and of order 2x2, and CB is also defined but is of order 3 x 3. Suppose that =

2 0 1 1

3 6 1 3

and = are not equal. We have EC =

; then and CB are both of order 2 x 2, but they

6 12 4 9

CB =

12 6 5 3

Because of all these complications, when we say that a matrix is multiplied by a matrix C, we have to specify whether is premultiplied by so that we have CB or whether it is postmultiplied by so that we have .

Reversal Law for Transpose of a Product

If and are matrices such that is defined, then ( ) = CB. This result can easily be verified and hence we shall not prove it.

Identity Matrix

Ann X n matrix with 1 in the diagonal and zeros elsewhere is called an identity matrix of order n and is denoted by I„. For example.

1 0 0 0 1 0 0 0 1

The identity matrix plays the same role in matrix multiplication as the number 1 in scalar multiplication, except that the order of I„ must be defined properly for pre- and postmultiplication; for example, if is 3 x 4, then I3B = and BI4 = B.

Inverse of a Matrix

The inverse of a square matrix A, denoted by A , is a matrix such that A~A = AA" = I. This is analogous to the result in ordinary scalar multiplication yy~ = >")= 1. To find A given A, we need to go through afew results on determinants.

Determinants

Corresponding to each sauare matrix A there is a scalar value known as the determinant, which is denoted by A. There are n elements in A, and the determinant of A is an algebraic function of these n elements. Formally, the definition is

\M = E (±)ai* 2* • • ««*

where the second subscript is a permutation of the numbers (1,2,. . . ,n). The summation is over all the (n!) permutations, and the sign of any element is positive if it is an even permutation and negative if it is an odd permutation.

The permutation is odd (even) if we need an odd (even) number of interchanges in the elements to arrive at the given permutation. For example, if we have a 3 X 3 matrix,

«21

LO31

«32

«23

a33J

We have to consider permutations of the numbers (1, 2, 3). The permutation (3, 2, 1) is odd because we need one interchange. The permutation (3, 1, 2) is even because we need two interchanges. There are (3!) = 6 permutations in all. These are, with the appropriate signs: +(1, 2, 3), -(1, 3, 2), -(2, 1, 3), + (2, 3, 1), -(3, 2, 1), +(3, 1, 2). Hence we have

A = ,022 - « - «12021033 + 12 23 31 - O13O22O31 + «13021032

Note that the first subscripts are always (1,2, 3). We can write this in terms of the elements of the first row as follows:

A = 0,1(022033 - O23O32) + Oi2(-02,033 + O23O31) + 0,3(02,032 - O22O31)

The terms in parentheses are called the cofactors of the respective elements. Thus if we denote the cofactor of a by „, of 0,2 by ,2, and of 0,3 by ,3, we have

A = „ „ + 0,21,2 + , ,

We can as well write A in terms of the elements and the corresponding cofactors of any other row or column. For example, if we consider the third column, we can write

A = , , + «2323 +

The cofactor is nothing but the determinant of a submatrix obtained by deleting that row and column, with an appropriate sign. For example, if we want to get the cofactor for 0,3, we delete the first row and third column, calculated the value of the determinant of the 2x2 matrix, and multiply it by (-1)+ = 1. To get the cofactor of O;, in a matrix A, we delete the /th row and jth column in A, compute jthe determinant, and multiply it by (- 1)+. For a 2 x 2 matrix A On 0,2

O21 O22

, we have A = «,,022 ~ 0,2021-

Properties of Determinants

There are several useful properties of determinants that can be derived by just considering the definition of a determinant.

1. If A* is a matrix obtained by interchanging any two rows or columns of A, then A* = -A. This is because all odd permutations become even and even permutations become odd-with one extra interchange. For example.

2. From property 1 it follows that if two rows (or columns) of a matrix A are identical, then A( = 0 because by interchanging these rows (or columns) the value is unaltered. Thus A = A* = -A. Hence A = 0. For example.

1 3 7 1 3 7 5 1 8

3. Expansion of a determinant by "alien" cofactors is equal to zero. By "alien" cofactors, we mean cofactors of another row or column. For example, consider the 3 x 3 matrix. The first row is (a,„ 0,2, Oo) with cofactors (A,„ A,2, , ). The second row is (021. «22, «23) with cofactors (A21, A22, A23). Then

« + aijAij + 1 , = expansion by own cofactors ! + ai2A22 + cin23 = expansion by alien cofactors

We know that the first expansion is A. The second expansion would be a correct expansion if a,, = «21. 12 = 022- and 0,3 = Uji, that is, if the first and second rows of A are identical. But we know from property 2 that in this case. A = 0. Hence we have

«11 21 + O12A22 + O13A23 = 0

or expansion by alien cofactors is zero.

4. The value of a determinant is unaltered by adding to any of its rows (or columns) any multiples of other rows (or columns). For example, consider adding three times the second row to the first row for a 3 x 3 matrix. We then have, expanding by the elements and cofactors of the first row.

«11 + 3a2i

«31

a,2 -b 322 «22

0,3 + 3023

«23

= (o„ + 3a2,)A„ + (0,2 + 3a22)A,2

+ (a,3 + 2 ) = ( ,, + ayA + , , ) + 3(a2iAi, + 022A,2 + )

The first term in parentheses is A, and the second is zero, because it is an expansion by alien cofactors.

This property is useful for evaluating the values of determinants. For instance, consider

iA =

2 3 -1

16 10 3

Now consider (row 1) - (row 2) - 2(row 3). The value of the determinant is unaltered. But we get

10 0 3 4 10 12 3

Now expand by the elements of the first row. It is easy because we have two zeros. We get