2. Some Properties of the Characteristic Polynomial of a Matrix and Their Use We consider a square matrix A 3 1 A = 1 2 with its characteristic polynomial o(r). Recall that according to the Cayley-Hamilton Theorem, o(A) = 0, where A is the eigenvalue of A. (a) Determine the eigenvalues of A and the polynomial o(x). (b) Consider another polynomial P(x) given by, P(x) = x+3³+2x2+x+1. If we write P(x)/o(x) = Q(r) + R(x)/o(x), identify Q(x) and R(x), where R(x) is a polynomial of order (n – 1), if A is n x n (in the present case n = 2). (c) Using (b), show that P(A) = R(A). On the other hand, P(A) = A + 3A³ + 2A² + A +I. Thus, this allows us to write any polynomial P(A) with any order >n in terms of a polynomial with order (n – 1).

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