3. Let A be a n x n matrix. Suppose that A 7 Onn, A # In, and A? = A. (A) Show that A has exactly 2 eigenvalues: 0, 1. (B) Let v E nullA. Show that v is an eigenvector of A with eigenvalue 0. (C), Let v E colA. Show that v is an eigenvector of A with eigenvalue 1. (D) Show that A is diagonalizable.

Let A = 3 1 2 4 Find the eigenvalues and the corresponding eigenvectors of matrix A ck that it is diagonalizable by computing AP and PD.

Let be an eigenvector of a matrix A corresponding to the eigenvalue 1, = -1 and the vector be an eigenvector of A corresponding to the eigenvalue 1, = 4. Which of the following is equal to A. e) L d-) e-) nndiğini görene kadar hekleviniz Soruvu bos birakmak isterseniz işarett

3. Let A be a nxn. matrix with distinct and positive eigenvalues. For each i, 1 ≤ i ≤n, let v; be an eigenvector of A with eigenvalue A, such that the v, are mutually orthogonal unit vectors. That is, (a) Suppose that w = j = 1,..., n. 1, for i = j, 0, for i j. av, for some a, E R. Prove that wv; = a; for all Vi Vj = (b) Show that x- (Ax) ≥ 0 for all x ER".