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Having a little a trouble with this one just some review I was doing on proofs, haven't done them in a while.
2026-04-02 07:31:23.1775115083
Elementary Linear Algebra Proof
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Let $A=\left[ \begin{matrix} a & b \\ c & d \end{matrix}\right]$. Then compute $AP$ and $PA$.
We get
$AP=PA \iff 2a+c=2a, 2b+d=a+3b, 3c=2c$ and $3d+c=3d$.
It is now easy to see that
$AP=PA \iff c=0$ and $d=a+b$.