Question

If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix X = [x, y, z] ≠ [0, 0, 0] such that

(A) PX = [0, 0, 0]

(B) PX = X

(C) PX = 2X

(D) PX = - X

Answer 1

Answer is (D) PX = −X

Given

PT = 2P + I

⇒ P = 2PT + I = 2(2P + I) + I

⇒ P + I = 0

⇒ PX + x = 0

⇒ PX = -X