The problem provides two matrix equations:

1. $2A - B = \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix}$
2. $A - 2B = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}$

We are tasked with finding $\det(AB^{-1})$.

Step 1: Solve for matrices $A$ and $B$

From the two equations, we can express both $A$ and $B$ in terms of these matrices. Let’s solve this system of equations for $A$ and $B$.

Equation 1:

$2A - B = \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix}$

Equation 2:

$A - 2B = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}$

Step 2: Find $\det(AB^{-1})$

Once we find $A$ and $B$, we will calculate $\det(A)$, $\det(B)$, and apply the property of determinants: $\det(AB^{-1}) = \frac{\det(A)}{\det(B)}$

Let me first calculate matrices $A$ and $B$.The solution for matrices $A$ and $B$ based on the system of equations is:

$A = \begin{bmatrix} -\frac{5}{3} & \frac{2}{3} \\ \frac{5}{3} & 4 \end{bmatrix}, \quad B = \begin{bmatrix} \frac{5}{3} & -\frac{5}{3} \\ -\frac{2}{3} & 1 \end{bmatrix}$

Step 3: Compute $\det(AB^{-1})$

Using the property of determinants, we know: $\det(AB^{-1}) = \frac{\det(A)}{\det(B)}$

Next, let's calculate the determinants of $A$ and $B$ and then find $\det(AB^{-1})$.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed with manual steps for calculation or explore another question?

Here are some related questions you might consider:

1. How do you calculate the determinant of a 2x2 matrix?
2. How is the inverse of a matrix calculated?
3. What properties of determinants are useful for simplifying complex matrix expressions?
4. How does matrix multiplication work in general?
5. What are some applications of matrix inverses in real-world problems?

Tip: When dealing with determinants of inverse matrices, remember that $\det(A^{-1}) = \frac{1}{\det(A)}$, which simplifies calculations involving matrix products.