Could you please help with question 159 and 160? Thanks

Convergence of matrix sequences Exercise 154 Convergence of matrix sums Exercise 155 Absolute convergence Exercise 156 Neuman n's Theorem Exercise 157 Exercise 158 38 3.7 Perturbed inverses We will now show that .A'1 is a continuous functions of its entries. There are several ways to establish this fact. We will take a route via Neumann's theorem that is useful in its own right. Let A\" for n = 1, 2, . . ., denote a sequence of m X n matrices. We say that 71 limnnDo A\" = A, if every component of A\" converges to the corresponding compo- nent of A. In other words convergence of a matrix sequence is dened component- wise. Show that lim\"_,,00 An = A ifE lim\"_,,00 IA\Exercise 159 Exercise 160 Exercise 161 Exercise 162 Exercise 163 Exercise 164 The only question is whether it converges to (I A) '1? First we prove the required inverse exists. Suppose it does not. Then there exists a vector x with "x" = 1 such that Ax = x. (My?) Show that this implies that "A" 2 1, which is a contradiction. It follows that I A is invertible. Suppose 230:1 A\" and 22:1 Bn are two absolutely converging matrix series. Show that ' 235:1 An + 220:1 Bu = 22:1(A + Bu) 0 C 230:1 An = 2:1 CA\" Show that (I A) 23;.) A\" = I. III Show that if A = 230:1 A\" then ||A| g 2::1 \"ARI. Show that if "A"