The Cayley-Hamilton Theorem: Let pA () = det(A - I) be the characteristic polynomial of A.

Question:

The Cayley-Hamilton Theorem: Let pA (λ) = det(A - λ I) be the characteristic polynomial of A.
(a) Prove that if D is a diagonal matrix, then pD(D) = O.
(b) Prove that if A is complete, then pA(A) = O.
(c) Prove that if J is a Jordan block, then pJ(J) = O.
(d) Prove that this also holds if J is a Jordan matrix.
(e) Prove that any square matrix satisfies its own characteristic equation: pA(A) = O.