Finding the inverse of a matrix from scratch.

Recall that, for a square n n matrix A, matrix B is the inverse of A if AB = BA = I_n.

Find the inverse of A = [3 4;5 6] by finding a 2 2 matrix B = [a b c d] for which AB =I₂ by following the steps below. Please follow the instructions carefully.

(a) Write down two systems (one in a and c, the other in b and d) corresponding to the matrix equation AB = I₂.

(b) Calculate the RREF required to solve these two systems (by hand - show your steps!), and find the solutions, thus finding the matrix B.

(c) How is this work related to row-reducing [A :I₂] ?