# Commutative 2x2 Matrices: Finding Solutions for AB = BA

• RJLiberator
In summary, the answer to this question is that you get the product of the matrices as follows:AB=BA\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}
RJLiberator
Gold Member

## Homework Statement

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

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

## 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.

RJLiberator said:

## Homework Statement

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

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

## 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}

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.

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

1. Is this correct?

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

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

$\begin{pmatrix} 0 & b \\ b & 0 \end{pmatrix}$ 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 $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

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?

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

[0bb0]​
\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

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.

RJLiberator
Thanks guys for the help here. Greatly appreciated.

## 1. What is a commutative 2x2 matrix?

A commutative 2x2 matrix is a square matrix with two rows and two columns where the order of multiplication does not affect the outcome. In other words, when two 2x2 matrices are multiplied in either order, the result will be the same.

## 2. How do you determine if a 2x2 matrix is commutative?

To determine if a 2x2 matrix is commutative, you can simply multiply the matrix by itself in both orders (AB and BA) and compare the results. If the result is the same, then the matrix is commutative.

## 3. Are all 2x2 matrices commutative?

No, not all 2x2 matrices are commutative. Only square matrices with equal dimensions can be commutative, and even then, it is not always the case.

## 4. What is the significance of commutativity in 2x2 matrices?

The commutative property in 2x2 matrices allows for simpler calculations and reduces the number of steps needed to perform operations. This makes it easier to solve equations and find solutions.

## 5. How is the commutative property used in real-life applications?

The commutative property of 2x2 matrices is used in various fields, such as physics, engineering, and computer graphics. It allows for efficient calculations and simplifies complex equations, making it an essential tool in problem-solving and data analysis.

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