b) If is an eigenvalue of an matrix A and x is a corresponding eigenvector, then 3 is an eigenvalue of A3 with corresponding eigenvector x. Solution A value is an eigenvalue of a matrix A with corresponding eigenvector x iff they satisfy the equation x = Ax. We are given this to start with. Left-multiply both sides by A: Ax = A2x. Scalar multiplication is commutative with matrix multiplication, so bring the out in front: Ax = A2x We know from our original equation that Ax = x, so substitute this in on the left: (x) = A2x 2x = A2x. Incidentally, this shows 2 is an eigenvalue for A2 with corresponding eigenvector x. We repeat, left-multiplying by A again: A2x = AA2x 2Ax = A3x Substitute in x for Ax on the left: 2(x) = A3x 3x = A3x showing 3 is an eigenvalue for A3 with corresponding eigenvector x..