Let {a1, .... an} and {b1, ..., bn} be two bases of ℝn. Let P be an n x n matrix with real entries such that Pai = bi i = 1, 2,..., n. Suppose that every eigenvalue of P is either −1 or 1. Let Q = I + 2P. Then which of the following statements are true?
1
{ai + 2bi | i = 1,2,..., n} is also a basis of V.
2
Q is invertible.
3
Every eigenvalue of Q is either 3 or -1
4
det Q > 0 if det P > 0.