The equation presented is a matrix equation of the form:

$\begin{pmatrix} 1 & -2 & -1 \\ -3 & 2 & 2 \\ 3 & -1 & -2 \end{pmatrix} \cdot X = \begin{pmatrix} 1 & 0 \\ 2 & -2 \\ -3 & 1 \end{pmatrix}$

To solve for the matrix $X$, we need to determine the values in $X$ that satisfy this equation. This can be done by either performing row operations to isolate $X$ or by finding the inverse of the left-side matrix (if it exists) and applying it to the right side.

Let me proceed with the calculation.The solution for the matrix $X$ that satisfies the given equation is:

$X = \begin{pmatrix} -2 & 4 \\ -1 & -1 \\ -1 & 6 \end{pmatrix}$

This matrix $X$ ensures that the product of the matrix on the left side and $X$ equals the matrix on the right side.

Would you like further details on this process or have any questions?

Here are some related questions you might consider:

1. How can you verify the solution by matrix multiplication?
2. What if matrix $A$ was not invertible? How would that affect the solution?
3. Could there be multiple solutions to this equation?
4. How does matrix inversion work, and what are its limitations?
5. What are some applications of solving matrix equations like this in real-world problems?

Tip: Always check the determinant of a matrix before attempting to find its inverse, as a zero determinant means the matrix is not invertible.