The system of matrix equations involves finding the unknown matrices $X$ and $Y$ using the given equations:

1. $-5X + 6Y = A$
2. $4X - 5Y = B$

Where matrices $A$ and $B$ are provided as: $A = \begin{pmatrix} 12 & -11 \\ 0 & -15 \\ 12 & 32 \end{pmatrix}, \quad B = \begin{pmatrix} -10 & 9 \\ 0 & 12 \\ -10 & -26 \end{pmatrix}$

Approach:

1. Solve the system for $X$ and $Y$ using linear combinations of matrices $A$ and $B$.
2. The form $X = c_1 A + c_2 B$ and $Y = d_1 A + d_2 B$ will help.
3. After determining $X$ and $Y$, calculate $X^T Y$ (where $X^T$ is the transpose of matrix $X$).
4. Finally, find the determinant of $X^T Y$.

Let me compute the solution step by step.The solution to the system of matrix equations is as follows:

1. The matrix $X$ is: 0 & 1 \\ 0 & 3 \\ 0 & -4 \end{pmatrix}$$
2. The matrix $Y$ is: 2 & -1 \\ 0 & 0 \\ 2 & 2 \end{pmatrix}$$
3. The product $X^T \cdot Y$ (where $X^T$ is the transpose of $X$) is: 0 & 0 \\ -6 & -9 \end{pmatrix}$$
4. The determinant of $X^T \cdot Y$ is: $\text{det}(X^T \cdot Y) = 0$

Would you like further details or clarifications?

Here are 5 related questions to expand the topic:

1. How do you find the transpose of a matrix?
2. What are the properties of matrix multiplication?
3. How is the determinant of a matrix calculated?
4. How do linear combinations of matrices work?
5. What happens when the determinant of a matrix is zero?

Tip: If the determinant of a matrix is zero, it means the matrix is singular and does not have an inverse.