- #1

RJLiberator

Gold Member

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## Homework Statement

Let A =

\begin{bmatrix}

0 & 1 \\

1 & 0

\end{bmatrix}

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

## Homework Equations

http://euclid.colorado.edu/~roymd/m3130/Exam2sol.pdf

## 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.