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To solve the given problem, we need to find \(\text{det}(AB^{-1})\). Here's a step-by-step approach:

**Step 1: Define Matrices A and B using the given equations.**

From the image:

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

**Step 2: Solve for A and B.**

We can rewrite these equations to express \(A\) and \(B\) in terms of known matrices.

From (Equation 1):

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

From (Equation 2):

\[
A = 2B + \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}
\]

We can solve these two equations simultaneously to find matrices \(A\) and \(B\). 

**Step 3: Find Matrix A and Matrix B.**

Substituting the value of \(B\) from Equation 1 into Equation 2:

\[
A - 2(2A - \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix}) = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}
\]

Simplifying:

\[
A - 4A + 2 \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix} = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}
\]

\[
-3A + 2 \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix} = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix}
\]

\[
-3A = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix} - 2 \begin{bmatrix} -5 & 3 \\ 4 & 7 \end{bmatrix}
\]

\[
-3A = \begin{bmatrix} -5 & 4 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} -10 & 6 \\ 8 & 14 \end{bmatrix}
\]

\[
-3A = \begin{bmatrix} 5 & -2 \\ -5 & -12 \end{bmatrix}
\]

\[
A = \begin{bmatrix} -\frac{5}{3} & \frac{2}{3} \\ \frac{5}{3} & 4 \end{bmatrix}
\]

We can find \(B\) using:

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

**Step 4: Find \( AB \).**

Calculate \(AB\) and then find \(\text{det}(AB)\). Finally, use the property of determinants \(\text{det}(AB^{-1}) = \frac{1}{\text{det}(AB)}\). 

Let's calculate the final result.The determinant of the matrix \( AB \) is approximately \(-4.321\).

Therefore, the determinant of \( AB^{-1} \) is approximately \(-0.231\).

Would you like a more detailed explanation or have any questions?

Here are some related questions to explore further:

1. How do you find the inverse of a matrix?
2. What are the properties of determinants in matrix multiplication?
3. How does the determinant change when scaling a matrix?
4. How do you solve systems of linear equations using matrices?
5. What is the significance of the determinant being zero?

**Tip:** Always verify the properties of matrix operations like multiplication and inversion to ensure accurate calculations!

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

Solve for the determinant of the inverse of the product of matrices A and B, given two matrix equations. This problem involves matrix multiplication, determinants, and matrix inversion, suitable for advanced high school or college-level students.

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