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Every square matrix satisfies its own characteristic equation.
ie: if A is a square matrix
and | xI − A | = f(x),
(so f(x) is the characteristic polynomial of A, and f(x)=0 is the characteristic equation of A),
then the Cayley-Hamilton theorem says that f(A) = 0.
eg: Let A = (

1	3
4	5

) , then
f(x) = | xI − A |

=    ( x −1) 3  4 ( x −5)   

= (x−1)(x−5) − 12

= x2 −6 x −7

Then, working in matrices rather than ordinary numbers (so we have 7.I instead of 7), f(A) = (

1	3
4	5

) ² − 6 (

1	3
4	5

) − (

7	0
0	7

)

=     13 18 24 37     −     6 18 24 30     −     7 0 0 7    

=     0 0 0 0    