Hey AuXian.

Firstly, if you have square matrices A and B then det(AB) = det(A)det(B) and P has to be square so if P^2 = P then det(P^2) = det(P)det(P) = det(P) which means that det(P) = 1 or 0. But if P has an inverse it can't equal 0 so it has to be 1.

Now for the second example, what is your distributive rule when it comes to matrices (i.e what is (A+B)^2 if it equals (A+B)(A+B))?