4. Suppose AE Rnxn with eigenvalue A. Given scalar a, prove that A+ a is an eigenvalue of A+ al, where I is the n × n identity matrix.

Determine all nn symmetric matrices that have 0 as their only eigenvalue.

Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has two distinct real eigenvalues, one real eigenvalue, and no real eigenvalues.

Show that A=[0110] has no real eigenvalues.