Let's break down the given problem and solve it step by step.

You are provided with two matrix equations:

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

You are asked to find the determinant of $\text{det}(AB^{-1})$.

Step-by-step approach:

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

   • From the first equation: $2A - B = \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix}$
   • From the second equation: $A - 2B = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}$

2. Find the determinant of $AB^{-1}$:

   • We will use the relationship: $\text{det}(AB^{-1}) = \frac{\text{det}(A)}{\text{det}(B)}$

I will first solve the system to find $A$ and $B$, then calculate the determinants.

Let me proceed with the solution.The matrices $A$ and $B$ are:

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

The determinants are:

$\text{det}(A) = -\frac{70}{9}, \quad \text{det}(B) = \frac{5}{9}$

Finally, the determinant of $AB^{-1}$ is:

$\text{det}(AB^{-1}) = \frac{\text{det}(A)}{\text{det}(B)} = -14$

Would you like further explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. What does the inverse of a matrix represent geometrically?
2. How is the determinant of a product of matrices related to the individual determinants?
3. What happens when the determinant of a matrix is zero?
4. How does matrix multiplication affect the size of matrices?
5. Why is the determinant of the inverse matrix the reciprocal of the original determinant?

Tip: Inverting matrices with determinants close to zero can lead to numerical instability, so always check the determinant before inverting!