The problem asks to find all matrices $X$ such that $AX = XA$ for the given matrix

$A = \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix}.$

This is a commutative property problem in matrix algebra, where we need to solve for $X$ such that it commutes with $A$.

Let $X = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}$.

Step 1: Compute $AX$ and $XA$

• Multiply $A$ and $X$: $AX = \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} -x_{11} + 3x_{21} & -x_{12} + 3x_{22} \\ x_{11} + 2x_{21} & x_{12} + 2x_{22} \end{pmatrix}.$

• Multiply $X$ and $A$: $XA = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \begin{pmatrix} -1 & 3 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} -x_{11} + x_{12} & 3x_{11} + 2x_{12} \\ -x_{21} + x_{22} & 3x_{21} + 2x_{22} \end{pmatrix}.$

Step 2: Equate $AX = XA$

Equating the components of $AX$ and $XA$, we obtain the following system of equations:

1. $-x_{11} + 3x_{21} = -x_{11} + x_{12}$,
2. $-x_{12} + 3x_{22} = 3x_{11} + 2x_{12}$,
3. $x_{11} + 2x_{21} = -x_{21} + x_{22}$,
4. $x_{12} + 2x_{22} = 3x_{21} + 2x_{22}$.

Step 3: Simplify the equations

• From Equation (1): $3x_{21} = x_{12}$, so $x_{12} = 3x_{21}$.

• From Equation (2): $-x_{12} + 3x_{22} = 3x_{11} + 2x_{12}$, substituting $x_{12} = 3x_{21}$, we get: $-3x_{21} + 3x_{22} = 3x_{11} + 6x_{21},$ which simplifies to $3x_{22} = 3x_{11} + 9x_{21}$, or $x_{22} = x_{11} + 3x_{21}$.

• From Equation (3): $x_{11} + 2x_{21} = -x_{21} + x_{22}$, substituting $x_{22} = x_{11} + 3x_{21}$, we get: $x_{11} + 2x_{21} = -x_{21} + x_{11} + 3x_{21},$ which simplifies to $x_{21} = x_{21}$ (no new information).

• From Equation (4): $x_{12} + 2x_{22} = 3x_{21} + 2x_{22}$, substituting $x_{12} = 3x_{21}$, we get: $3x_{21} + 2x_{22} = 3x_{21} + 2x_{22},$ which is an identity (no new information).

Step 4: General form of $X$

From the above, $x_{12} = 3x_{21}$ and $x_{22} = x_{11} + 3x_{21}$. Thus, $X$ can be written as: $X = \begin{pmatrix} x_{11} & 3x_{21} \\ x_{21} & x_{11} + 3x_{21} \end{pmatrix},$ where $x_{11}$ and $x_{21}$ are free parameters.

Final Answer:

All matrices $X$ that satisfy $AX = XA$ are of the form: $X = \begin{pmatrix} a & 3b \\ b & a + 3b \end{pmatrix},$ where $a, b \in \mathbb{R}$ are arbitrary real numbers.

Would you like a more detailed breakdown of any part or further assistance? Here are some follow-up questions:

1. What are the properties of matrices that commute with a given matrix?
2. How does this solution change if $A$ has distinct eigenvalues?
3. What is the geometric interpretation of the solution space for $X$?
4. Can this commutativity condition be extended to higher dimensions?
5. How would the solution change if $A$ were diagonalizable?

Tip: When solving commutative matrix problems, analyzing eigenvalues and eigenspaces of $A$ can often simplify the work.