Is S Closed Under Addition in R^(2x2)?

[WTFsandwich]

Homework Statement

Suppose A is a vector $$\in$$ $$R^{2x2}$$.

Find whether the following set is a subspace of $$R^{2x2}$$.

$$S_{1} = {B \in R^{2x2} | AB = BA}$$

The Attempt at a Solution

I know that S isn't empty, because the 2 x 2 Identity matrix is contained in S.

The problem I'm having comes in the proof that addition is closed.

If I show A(B + C) = (B + C)A that should be sufficient, right?

So far I have:

Suppose $$B$$ and $$C$$ $$\in$$ S.
$$A(B + C) = (B + C)A$$
$$AB + AC = BA + CA$$

And that's where I'm stuck. I have no idea where to continue on to. Any help would be greatly appreciated.

 

[Mark44]

You need to show that S is closed under addition and scalar multiplication. You probably want to do those separately.

 

[WTFsandwich]

I know that's what I have to do, but I don't know how to go about doing it. I started the addition part up above, and am stuck at that point.

 

[Mark44]

OK, B and C are both elements of S.
A(B + C) = AB + AC (since vector multiplication is left-distributive)
Now, what can you say about AB and AC, since B and C are members of set S?