Cayley-Hamilton Theorem

Given

(1)

(2)

then

(3)

where  is the identity matrix. Cayley verified this identity for  and 3 and postulated that it was true for all . For , direct verification gives

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

so

(12)

The Cayley-Hamilton theorem states that an  matrix  is annihilated by its characteristic polynomial , which is monic of degree .