Take a polynomial
.
If *A* represents a square matrix of order *m*,
we define *P*(*A*) in the following way:

where *I* represents the identity matrix of order *m*.

Note that
*a*_{0}=*P*(0)=|*A*|.

We can use the theorem of Cayley-Hamilton to compute the multiplicative inverse of a square matrix, if it exists:

Recall that *A* is invertible if, and only if,
(v.s. 1.1) and suppose that
*A* is invertible (i.e.
,
i.e.
). Then:

i.e.

.