Let 1 A = 1 0 1 02 2 (a) Find an ordered basis for ran(A) (b) Perform the Gram-Schmidt algorithm on the basis you got in (a) to get an orthonormal basis for ran(A) (c) Find a basis for ker(A)

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

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

Q2) Let 23 - 222 – 32 be the characteristic polynomial of a matrix A. (i) Determine the eigenvalues of A. (ii) Is A invertible? Justify. (iii) What is the rank for 2I - A for each eigenvalue of A. | (ii) Find 4 and trace(A²).