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 a0=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: