Consider the matrix A = substituting z = A: (8) 0 Define an "evaluation" homomorphism: R[x] - → M₂ (R) by (anx + an-1xn-1. +...+ a₁x+ao) Here the powers A denote the usual powers of the matrix A under matrix multiplication, and I = denotes the identity matrix. (² 0 You do not have to prove that is a homomorphism, but I encourage you to think about why this is true. (a) Compute A². What is Ak for k ≥ 3? (b) Show that the image of the homomorphism is: = an A" + an-1A−¹ +...+ a₁A+aoI ao 0 b-{(2) man} Im(x) = ao, a₁ € R R} a1 ao (c) The characteristic polynomial of A is g(x) = x² of A. Show that g € Ker(p). (d) Issurjective? Injective? 0 1