Are the following statements true or false? If true, prove the statement. If false, give a counterexample.

1. Let A R 33 . If each eigenvalue of A is zero, then A2 = 0. (Note: 0 denotes the zero matrix R 33 )

2. Let A R nn be a real symmetric matrix, and 1, 2, ..., n the eigenvalues of A, then A2 = max i |i | where |i | denotes the absolute value of i for i = 1, ..., n.

3. Let A R nn and aij denote the element in i th row and j th column. If 0 aij < 1 i {1, ..., n}, j {1, ..., n}, and Pn j=1 aij < 1 for i = 1, ..., n, then (A) < 1 where (A) is the spectral radius of A.