Question

Let A be a 2 × 2 matrix satisfying A^k = 0 for some positive integer k. Show that A^2 = 0.

Answer 1

Suppose that λ is an eigenvalue of A with corresponding eigenvector v. Now, since Av = λv, Av = λkv. Since A^k = 0,therefore, λkv = 0. Since v is a non-zero vector (by definition of eigenvector), we conclude that λk = 0.

Thus λ = 0 ( as k being a positive integer, is non-zero).Thus, the only eigenvalue of A is 0. HYence the chareacteristic equation of A is (ʎ-0)(ʎ-0)= 0 or, ʎ² = 0.Now, as oper the Cayley Hamilton theorem, A satisfies its own chareacteristic equation, so that A² = 0.