1) For a matrix B to be the inverse of matrix A, it must satisfy the following equation:

So basically, if you can make up a random non-singular matrix and call it A, then B will be its inverse, and I'm sure you know how to find that!

2)

So now you have to find a matrix A whose inverse is also its square!

Hence:

, etc. 4 equations, 4 unknowns, you do the math.