Page 159 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Consider the matrices

We leave it for the reader to verify that

Thus , as guaranteed by Theorem 2.3.4.

The following theorem gives a useful relationship between the determinant of an invertible matrix and the determinant of its
inverse.

THEOREM 2.3.5

If A is invertible, then

Proof Since , it follows that . Therefore, we must have . Since ,
.
the proof can be completed by dividing through by

Linear Systems of the Form

Many applications of linear algebra are concerned with systems of linear equations in unknowns that are expressed in the
form

(6)

where is a scalar. Such systems are really homogeneous linear systems in disguise, since 6 can be rewritten as
or, by inserting an identity matrix and factoring, as

(7)

Here is an example:

EXAMPLE 5 Finding
The linear system
can be written in matrix form as
which is of form 6 with