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The problem is:

"Find all $2\times 2$ matrices A that have the property that for any $2\times 2$ matrix B, AB = BA."

Given hint: "The given equation must hold for all B. Try matrices B that have lots of zero entries."

I tried bashing by letting $A=\begin{pmatrix}a & b\\c & d\end{pmatrix}$ and $B=\begin{pmatrix}w & x\\y & z\end{pmatrix}$, but as you can tell, this got messy quickly. Any help would be appreciated, thank you.

Andrew Wang
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1 Answers1

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$$\begin{pmatrix}a & b\\c & d\end{pmatrix} \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}=\begin{pmatrix}a & 0\\c & 0\end{pmatrix}$$

$$\begin{pmatrix}1 & 0\\0 & 0\end{pmatrix} \begin{pmatrix}a & b\\c & d\end{pmatrix}=\begin{pmatrix}a & b\\0 & 0\end{pmatrix}$$

So, you get $b=c=0.$

$$\begin{pmatrix}a & b\\c & d\end{pmatrix} \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix}=\begin{pmatrix}0 & a\\0 & c\end{pmatrix}$$

$$\begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \begin{pmatrix}a & b\\c & d\end{pmatrix}=\begin{pmatrix}c & d\\0 & 0\end{pmatrix}$$

So, you get $c=0, a=d.$

That is,

$$A=\begin{pmatrix}a & 0\\0 & a\end{pmatrix}. $$ You only need to show that this matrices commute with any $2\times 2$ matrix.

mfl
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