Let A be a diagonalizable matrix with characteristic polynomial (a) If D is a diagonal matrix whose

Question:

Let A be a diagonalizable matrix with characteristic polynomial

(a) If D is a diagonal matrix whose diagonal entries are the eigenvalues of A, show that
= p(D) + a1Dn + a2Dn-1 + an+1I = 0
(b) Show that p(A) = O.
(c) Show that if a+ 0 ‰ , then A is nonsingular and A-1 = q(A) for some polynomial a of degree less than n.