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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!

Find the determinant of AB^-1, given 2A - B = [[-5, 3], [4, 7]] and A - 2B = [[-5, 4], [3, 2]]

Solve the determinant of AB^-1 from two matrix equations, with detailed steps on how to find A and B. This problem covers matrix inversion, determinants, and algebraic manipulation. Suitable for high school students in Grades 11-12.

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