5. (8 marks) Let A and B be n x n matrices. (a) Prove that if B is invertible, then det(B-¹AB) = det(A). (b) If A² = A (i.e. A is idempotent), find all possible values of det(A). (c) Prove that the only invertible idempotent matrix is the identity matrix. (d) Prove that if AB is invertible, then both A and B are invertible.