Find the steady-state vector for the matrix below. 0.3 0.5 0.5 P= 0.2 0 0.2 0.5 0.5 0.3 (Type an integer or simplified fraction for each matrix element.)

Consider the following matrix. P =    0.1 0.1   0.9 0.9 (a) Calculate P2 = P · P.                             (b) Calculate P4 = P2 · P2.                             (c) Calculate P8.                             (d) Without computing it explicitly, find P1,000.

Q 3.9 You are given the following matrix A = find a matrix (W) that diagonalizes the given matrix (A).

Let P be a 6x3 matrix and Q be a 3x6 matrix. Also, let A and B be linear functions A(x)=P(x) and B(y)=Q(y), where x,y are vectors. Fill in the blanks of the question below and explain your answers. 1. A is a function from Rs to Rt, where s=____ and t=____. The matrix of the composition BA equals ____. The null space of P is a subspace of Ra, where a=____. The range of P is a subspace of Rb where b=____. The largest possible value of the rank(Q) is ____. The smallest possible value of nullity(Q) is ____. Please indicate any definitions, diagrams or theorems that were used to help solve the problem.