5.12. The Determinant of the Inverse of a Nonsingular Matrix

We recall that a square matrix A is nonsingular if it has a left inverse B such that BA=I .  It is then shown that B, if exists, is unique and also a right inverse, i.e., AB=I .  Therefore, we call B the inverse of A and denote it by A 1 .

Theorem 5.12.

If matrix A is nonsingular, then detA0  and

det( A 1 )= 1 detA                                      (5.16)

Proof

Using the product formula (5.15), we have

        det( A A 1 )=( detA )( det A 1 )   =detI=1                 QED.

Comment

Theorem 5.12 shows that detA0  is a necessary condition for A to be nonsingular.  Later, in Theorem 5.16, it will be shown that it is also a sufficient condition, i.e., detA0  implies the existence of A 1 .