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# The theorem of Cayley-Hamilton.

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|.

Theorem 11.2.1 (Cayley-Hamilton)   If is the characteristic polynomial of the square matrix A, then P(A)=0.

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.

.

Noah Dana-Picard
2001-02-26