18. Diagonalize the matrix A = (hint: first take the fraction inside the matrix), and hence find lim A". 818 19. (a) Show by matrix multiplication that (I+ A+ A++ A")(IA) = 1 - An+1 for any square matrix A. Deduce that if A" O as noo, then I+ A+ A++A" (IA). (Compare the geometric series formula.) (b) In a recycle chemical reactor involving k reagents, the quantities of reagents present can be considered to form a vector in Rk. After n recycles the vector is denoted Xn, and it can be shown that X = Xo + AXn-1, where Xo is the input vector and A is a fixed matrix representing the reaction and the removal of finished product. (i) Show that X = (I+ A+ A++ A")Xo. (Hint: use induction.) (ii) By letting noo in (i), and using part (a), show that if all the eigenvalues of A have modulus less than 1, then X, approaches a steady state vector X, which is given by the formula X = (I-A)-X0.