# Commutative 2x2 Matrices

1. Sep 5, 2015

### RJLiberator

1. The problem statement, all variables and given/known data

Let A =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}

Find all 2 x 2 matrices B such that AB = BA.

2. Relevant equations

3. The attempt at a solution

I let B =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} and set AB=BA.

From here I see that a and d must be 0, and b=c must be true.

So the answer will be that all matrices that are commutative will be of form:

\begin{bmatrix}
0 & b \\
b & 0
\end{bmatrix}

And there is no other possible commutative matrix outside of this form.

1. Is this correct?
2. Is there any further proof of this needed?

Thank you kindly.

2. Sep 5, 2015

### pasmith

That must be wrong, because the 2x2 identity matrix commutes with every 2x2 matrix but is not of that form.

What did you actually get for AB and BA? I would suggest double-checking those calculations.

You have the right idea, but have not executed it correctly.

3. Sep 5, 2015

### RJLiberator

So to get this right:

if A = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}

All possible commutative matrices with matrix A should be in the form:

\begin{bmatrix}
0 & b \\
b & 0
\end{bmatrix}

This is the wrong answer? It seems to be right when I calculate it. I get the same answer either way. AB = BA

4. Sep 5, 2015

### pasmith

$\begin{pmatrix} 0 & b \\ b & 0 \end{pmatrix}$ is a subset of the matrices you are looking for. It can't be all of them, because the identity $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ commutes with every 2x2 matrix.

Recheck your initial calculations with $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

5. Sep 5, 2015

### RJLiberator

Ahhh, I see.

I calculated it out and found that while b=c, also a=d:

\begin{bmatrix}
a & b \\
b & a
\end{bmatrix}

This makes sense to me as the original answer was a subset of this.

Is there any further proof needed to show that is all?

6. Sep 5, 2015

### SammyS

Staff Emeritus

What do you get for the product $\displaystyle \ \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix} \ \ \ ?$

What do you get for the product $\displaystyle \ \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \ \ \ ?$

(I see you posted you answer just before I posted this.)

That looks good.

7. Sep 5, 2015

### RJLiberator

Thanks guys for the help here. Greatly appreciated.