A couple of questions...

[isuckatmath]

I am having some trouble with the following two questions. Any help would be greatly appreciated!

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.
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.

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

[stapel_eliz]

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.

If Ax₀ = 0 and Ax₁ = b, then what is the result of A(x₀ + x₁)? (Hint: Multiplication distributes over addition.)

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.

Do the same distribution thing.

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

Since Ax = x, then Ax - x = Ax - Ix = 0. Now do the reverse of distribution, and see what you get.

[isuckatmath]

omg that was so much easier than i was making it out to be. Thanks so much!!!!!