(1) If A and B are positive semidefinite matrices, then the eigenvalues of A.B are all nonnegative. 1

If A is a 3 x 3 matrix which is diagonalizable and the following. three vectors are eigenvectors for 3 distinct eigenvalues of A, (0, 1, –1)", (2, 0, 3)7, (1, 1, -2)7, which of the following vectors could be the first column of a matrix P which diagonalizes A? A) (1, 1, -2)7 (B) (0, 0, 1)7 (C) (1, 0, 0)7 D) (0, 1, 0) (E) (-1,0, –1)7 (F) (1, 2, -3)7-(G) (0, 1, 1)7 H) (2, 0, -3)7

7. Determine all 2 × 2 matrices A with eigenvalues 1 and –1 and corresponding eigenvectors and respectively.

Consider the matrix 2 2 A = 4 2 2 (a) Compute the eigenvalues and eigenvectors of A. (b) Find rank one matrices B and C such that A = B + C and BC 0, explaining why the matrices you found have these properties. (hint: use your answer to part (a))