Take a square matrix *A* of order *n* and a polynomial *T*(*x*) of degree *r*, such that *r*>*n*.
How can we compute *T*(*A*)?

Of course a direct computation is always possible, but perhaps not so illuminating.

Denote by *P*_{A}(*x*) the characteristic polynomial of *A* and then use Euclide's algorithm: there exists a unique ordered pair of polynomials
(*Q*(*x*),*R*(*x*)) such that
*T*(*x*)=*Q*(*x*) *P*_{A}(*x*) +*R*(*x*) and
.

By Cayley-Hamilton's Theorem (v.s. 2.1), we have:

.