[Math] How to find all the matrices that commute with this matrix

Question

I would like to find all the matrices that commute with the following matrix

$$A = \begin{pmatrix}2&0&0\\ \:0&2&0\\ \:0&0&3\end{pmatrix}$$

I set $AX = XA$, but still can't find the solutions from the equations.

Answer 1

$AX$ doubles the first two rows of $X$, and triples the third row.

$XA$ doubles the first two columns of $X$, and triples the third column. These two must agree.

This gives us a matrix $$X=\left(\begin{matrix}*& * & 0\\*&*&0\\0&0&*\end{matrix}\right)$$

Where $*$ can be anything.