Showing that the inverse is given by that particular matrix (with ad-bc in the denominators) gives you one direction. To do the other, suppose ad-bc=0 and try multiplying the top row by d and the bottom row by b.
Show that the matrix is nonsingular if and only if ad-bc≠0. If this condition holds, show that
I think what I want to do is show that is row equivalent to . When I was doing this I got stuck at