The problem asks to find all matrices $X$ such that $AX = XA$, where $A = \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix}$.

This is a commutativity condition, and we need to determine the set of matrices $X$ that commute with $A$.

Solution Outline

Let $X = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}$. We need to solve for the entries $x_{11}, x_{12}, x_{21},$ and $x_{22}$ such that $AX = XA$.

1. Calculate $AX$: $AX = \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} -x_{11} + 3x_{21} & -x_{12} + 3x_{22} \\ x_{11} + 2x_{21} & x_{12} + 2x_{22} \end{pmatrix}$

2. Calculate $XA$: $XA = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} -x_{11} + x_{12} & 3x_{11} + 2x_{12} \\ -x_{21} + x_{22} & 3x_{21} + 2x_{22} \end{pmatrix}$

3. Set $AX = XA$ and equate corresponding elements: From the matrix equality $AX = XA$, we get the following system of equations: $-x_{11} + 3x_{21} = -x_{11} + x_{12}$ [ -x_{12} + 3x_{22