If Ax0 = 0 and Ax1 = b, then what is the result of A(x0 + x1)? (Hint: Multiplication distributes over addition.)The first question:
a. Use matrix multiplication to show that if x0 is a solution of the homogeneous system Ax = 0 and x1 is a solution of the nonhomogeneous system Ax = b, then x0 + x1 is also a solution of the nonhomogeneous system.
Do the same distribution thing.b. Suppose that x1 and x2 are solutions of the nonhomogeneous system of part (a). Show that x1 - x2 is a solution of the homogeneous system Ax = 0.
Since Ax = x, then Ax - x = Ax - Ix = 0. Now do the reverse of distribution, and see what you get.The second question:
Let A be an n x n matrix such that Ax = x for every n-vector x. Show that A = I.