Let's start by solving the problem step by step.

Given:

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

We need to find $\det\left((AB)^{-1}\right)$.

Step 1: Express $A$ and $B$ in terms of matrices

First, let's express $A$ and $B$ in terms of these equations.

We have two equations:

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

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

Step 2: Solve the system for $A$ and $B$

Let's solve for $B$ from the first equation:

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

Substitute $B$ into the second equation:

[ A - 2\left(2A + \begin{pmatrix} 5 &