# Matrix algebra proof

1. Mar 9, 2015

### Kavorka

This problem is so simple that I'm not exactly sure what they want you to do:

Let A and B be n x n matrices such that AB = BA. Show that (A + B)^2 = A^2 + 2AB + B^2. Conclude that (I + A)^2 = I + 2A + A^2.

We don't need to list properties or anything, just manipulate. This all seems self-evident from the distributive property, and showing that I^2 = I.

2. Mar 9, 2015

### jfizzix

If AB = BA
and (A+B)^2 = (A+B)(A+B)
then the rest more or less falls into place.

3. Mar 9, 2015

### Kavorka

So AB = BA
(A+B)(A+B) = A^2 + 2AB + B^2
AI=IA=A
II = I
(I+A)(I+A) = I^2 + 2AI + A^2 = I + 2A + A^2

Would this probably be what they're looking for? Not sure how much more in detail I can go

4. Mar 10, 2015

### Staff: Mentor

I think you need some more detail here. Is it important that AB = BA in your proof?
I think you need some more detail here as well, particularly in how you expand (I + A)(I + A).

5. Mar 10, 2015

### WWGD

First of all, great name. Then, like Mark said, just expand the product term-by-term, without grouping.