Gaussian elimination 221

Consequences for Matrix Theory

Any system of linear equations can be represented via a

coefficient

MATRIX

and a column of constant values.

For instance, for our example:

y+ 3z= 0

2x+ 4y– 2z= 18

x+ 5y+ 3z= 14

the coefficient matrix, call it A, is given by:

the column of constant values, call it b, is:

Any elementary row operation performed on the origi-

nal set of equations corresponds to an operation on the

rows of the coefficient matrix Aand the column matrix

b. For instance, in the example above, our first opera-

tion was to interchange the first and second rows. This

can be accomplished by multiplying Aand beach by

the

PERMUTATION

matrix

We have:

Similarly, the elementary row operation of dividing the

first row through by 2 is accomplished by multiplica-

tion with the matrix

and the act of subtracting the first equation from the

first is accomplished by multiplication with the matrix:

In this way one can see that every elementary row oper-

ation corresponds to multiplication by an elementary

matrix. This observation has an important consequence.

The inverse of a square matrix Ais simply the

product of the elementary matrices that reduce

Ato the identity matrix.

Our example explains this. We used elementary row

operations to reduce the system of equations to the

equivalent system:

x= 2

y= 3

z= –1

That is, we found a collection of eight elementary

matrices E1, E2, …, E8such that application of these

eight matrices reduced the matrix of coefficients Ato

the

IDENTITY MATRIX

I.

E8E7E6E5E4E3E2E1A= I

If we let Bbe the matrix E8E7E6E5E4E3E2E1, then

we have BA = I, which means that B= A–1, the

INVERSE

MATRIX

to A. (As the

DETERMINANT

of the identity

matrix Iis 1, the equation E8E7E6E5E4E3E2E1A= I

shows that the determinant of Acannot be zero. Thus,

as the study of determinants shows, the matrix Adoes

indeed have an inverse.) Notice that the matrix Bis the

same elementary row operations applied to the matrix

I. This result provides a constructive method for com-

puting the inverse to a matrix A.

To compute the inverse of a matrix A, write the

matrix Aand the matrix Iside by side. Perform