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The problem involves solving for matrices \( A \) and \( B \) given two matrix equations and then calculating \( \det(AB^{-1}) \).

### Given:
1. \( 2A - B = \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix} \)
2. \( A - 2B = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix} \)

We need to find the determinant of \( AB^{-1} \).

### Steps to solve the problem:

1. **Solve for \( A \) and \( B \):** Use the two matrix equations to isolate \( A \) and \( B \).
2. **Find the inverse of \( B \):** Once we have \( B \), calculate \( B^{-1} \).
3. **Calculate the product \( AB^{-1} \):** Multiply \( A \) by the inverse of \( B \).
4. **Find the determinant of \( AB^{-1} \):** Use the properties of determinants to find \( \det(AB^{-1}) \).

Let me calculate this step-by-step.The solution to the problem is as follows:

1. **Matrix \( A \):**
   \[
   A = \begin{bmatrix} -\frac{5}{3} & \frac{2}{3} \\ \frac{5}{3} & 4 \end{bmatrix} 
   \approx \begin{bmatrix} -1.67 & 0.67 \\ 1.67 & 4 \end{bmatrix}
   \]

2. **Matrix \( B \):**
   \[
   B = \begin{bmatrix} \frac{5}{3} & -\frac{5}{3} \\ -\frac{2}{3} & 1 \end{bmatrix}
   \approx \begin{bmatrix} 1.67 & -1.67 \\ -0.67 & 1 \end{bmatrix}
   \]

3. **Inverse of matrix \( B \) (\( B^{-1} \)):**
   \[
   B^{-1} = \begin{bmatrix} 1.8 & 3 \\ 1.2 & 3 \end{bmatrix}
   \]

4. **Matrix product \( AB^{-1} \):**
   \[
   AB^{-1} = \begin{bmatrix} -2.2 & -3 \\ 7.8 & 17 \end{bmatrix}
   \]

5. **Determinant of \( AB^{-1} \):**
   \[
   \det(AB^{-1}) \approx -14
   \]

The determinant of \( AB^{-1} \) is \( \boxed{-14} \).

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Do you have any questions or would you like further details on any part of the solution?

Here are 5 related questions you could explore:
1. How can we solve similar matrix equations with larger dimensions?
2. How does the determinant of a matrix product relate to the determinants of the individual matrices?
3. What is the geometric interpretation of a matrix inverse?
4. How do we calculate the determinant of a 3x3 or higher-dimensional matrix?
5. What are the conditions for a matrix to be invertible?

**Tip:** Always verify the existence of an inverse by checking if the determinant of the matrix is non-zero before attempting to compute the inverse.

Find det(AB^{-1}) given the matrix equations 2A - B = [[-5, 3], [4, 7]] and A - 2B = [[-5, 4], [3, 2]].

This problem involves solving for matrices A and B given two matrix equations and calculating det(AB^{-1}). The solution includes matrix algebra, matrix inversion, and the determinant formula for 2x2 matrices. Suitable for students at the undergraduate level studying linear algebra.

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