Find eta and ak for any integer - so do not compute the

Question

For the matrices A in problems 1 and 2, find e^tA and A^k for any integer k ≥ 0.

Problem 1: A =

Problem 2: A =

Problem 3: Using the same method as in class, define (I - A)^-1  by the power series representation (I - A)^-1 = ∑_k=0^∞ A^k (assuming that (I - A^-1) commutes with A and then find (I - A)^-1 for A =

(The point is that the method works for any convergent power series, so do not compute the result using the MATH 220 way to find inverses of 2x2 matrices.)