Matrix Polynomials. Let p(x) = cnxn+cn-1 xn-1+ + C1x + C0 be a polynomial function. If A is a square matrix, we

Matrix Polynomials. Let p(x) = cnxn+cn-1 xn-1+ ˆ™ ˆ™ ˆ™ + C1x + C0 be a polynomial function. If A is a square matrix, we define the corresponding matrix polynomial p(A) = cnAn + cn-1An-1 + ˆ™ ˆ™ ˆ™+ C1A + c0I; the constant term becomes a scalar multiple of the identity matrix. For instance, if p(x) = x2 - 2x + 3, then p(A) = A2 - 2A + 3 I.
(a) Write out the matrix polynomials p(A), q(A) when p(x) = x3 - 3x 4- 2, q(.x) = 2.x2 + 1.
(b) Evaluate p (A) and q (A) when

(c) Show that the matrix product p(A)q(A) is the matrix polynomial corresponding to the product polynomial r(x) = p(x)q(x).
(d) True or false; If B = p(A) and C = q(A) then BC = CB. Check your answer in the particular case of part (b).

A 11)