Recall that if p = C[t] and p(t) = adtd + ad1td1 + ... + art...

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Recall that if p = C[t] and p(t) = adtd + ad−1td−1 + ... + art + ao, then for A Є M, (C) we define p(A) = adAd + ad-1Ad-1 + ... + a₁A + aoln. For this problem, fix pЄ C[t] and A Є Mn(C). (i) If A is an eigenvalue of A, show that p(A) is an eigenvalue of p(A) and that the geometric multiplicity of p(A) in p(A) is at least the geometric multiplicity of A in A. (ii) In the previous part, does equality hold? Either provide a proof that it does, or prove that there exists a counterexample. (iii) Does the algebraic multiplicity of A in A equal in the algebraic multiplicity of P(A) in p(A)? Prove this is true, or prove that there exists a counterexample. Recall that if p = C[t] and p(t) = adtd + ad−1td−1 + ... + art + ao, then for A Є M, (C) we define p(A) = adAd + ad-1Ad-1 + ... + a₁A + aoln. For this problem, fix pЄ C[t] and A Є Mn(C). (i) If A is an eigenvalue of A, show that p(A) is an eigenvalue of p(A) and that the geometric multiplicity of p(A) in p(A) is at least the geometric multiplicity of A in A. (ii) In the previous part, does equality hold? Either provide a proof that it does, or prove that there exists a counterexample. (iii) Does the algebraic multiplicity of A in A equal in the algebraic multiplicity of P(A) in p(A)? Prove this is true, or prove that there exists a counterexample.