This matrix equation appears to involve solving for matrix $X$. The general form of the equation is:

$\begin{bmatrix} 1 & -4 \\ 5 & -19 \end{bmatrix} X = \begin{bmatrix} 3 & 3 \\ -3 & -6 \end{bmatrix}$ and $X = \begin{bmatrix} -69 & -81 \\ 12 & 9 \end{bmatrix}$

To solve this:

1. Multiply the first matrix on the left-hand side by $X$ to express the system in terms of the unknown matrix.
2. Set the resulting matrix equal to the right-hand matrix, $\begin{bmatrix} 3 & 3 \\ -3 & -6 \end{bmatrix}$.
3. Then substitute the known value of $X$ into this equation to verify if it satisfies the original equation.

Would you like a detailed step-by-step solution for this matrix multiplication and verification?

Here are 5 related questions to extend this:

1. How do we solve matrix equations in general?
2. What are some methods to verify solutions of matrix equations?
3. Can matrix multiplication be done element-wise?
4. What is the significance of matrix inverses in solving such systems?
5. How would this problem change if one of the matrices was singular?

Tip: Always check the determinant of square matrices when dealing with inverses; a zero determinant indicates that the matrix is singular and not invertible.