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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.

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