The first question:

a. Use matrix multiplication to show that if x

_{0}is a solution of the homogeneous system Ax = 0 and x

_{1}is a solution of the nonhomogeneous system Ax = b, then x

_{0}+ x

_{1}is also a solution of the nonhomogeneous system.

b. Suppose that x

_{1}and x

_{2}are solutions of the nonhomogeneous system of part (a). Show that x

_{1}- x

_{2}is a solution of the homogeneous system Ax = 0.

The second question:

Let A be an n x n matrix such that Ax = x for every n-vector x. Show that A = I.