## 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
detA≠0
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
detA≠0
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.,
detA≠0
implies the existence of
A
−1
.