Prove that if A2 = O, then 0 is the only eigenvalue of A. STEP 1: We need to show that if there exists a nonzero vector x and a real number λ such that Ax = λx, then if A2 = O, λ must be . STEP 2: Because A2 = A · A, we can write A2x as A(Ax). STEP 3: Use the fact that Ax = λx and the properties of matrix multiplication to rewrite A2x in terms of λ and x. A2x = x STEP 4: Because A2 is a zero matrix, we can conclude that λ must be .
