2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 , then define p(A) = E a¡A' .If,A is an eigenvalue of A, the show that p() is an eigenvalue of p(A). i = 1

Suppose that a matrix A has the following eigenvalues and eigenvectors: • A = -1 has eigenvector x = [1, 2, 3]T. •A = 2 has eigenvector y = [-1, 0, 2]". •A = 3 has eigenvector z = [0, 2, 1]". (i) Write the vector v = [1, 10, 11]T as a linear combination of x, y, and z. (ii) Calculate Av using your solution from Part (i).

Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2.

Find a 3 x 3 matrix A (with real number entries) whose characteristic polynomial is det(A – zls): * + 2 + 3r +9. A =