The given problem is a matrix equation:

$X(A^{-1}X)^{-1}X - 3X = BCX^{-1}(B^{-1}X^{-1})^{-1}$

where the matrices are:

$A = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$

Would you like me to solve for $X$ step by step? Let me know if you want an explanation or further clarification.

Here are five follow-up questions that expand on this topic:

1. How do you compute the inverse of a 2×2 matrix?
2. What are the properties of matrix inverses that can simplify this equation?
3. How can this equation be interpreted geometrically?
4. What numerical methods can be used to solve matrix equations when an analytical solution is difficult?
5. How does this equation change if we assume $X$ is diagonal?

Tip: When solving matrix equations, always check if the given matrices are invertible before attempting inversion.