Hamilton-Cayley theorem | matrix polynomial identity

Given A | = | left bracketing bar a_11 - x | a_12 | ... | a_(1m) a_21 | a_22 - x | ... | a_(2m) ⋮ | ⋮ | ⋱ | ⋮ a_(m1) | a_(m2) | ... | a_(m m) - x right bracketing bar | = | x^m + c_(m - 1) x^(m - 1) + ... + c_0, then A^m + c_(m - 1) A^(m - 1) + ... + c_0 I = 0, where I is the identity matrix. Cayley verified this identity for m = 2 and 3 and postulated that it was true for all m.