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

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 &

Explore how to solve matrix equations involving determinants and inverses with detailed step-by-step solutions. This problem covers matrix algebra concepts, including multiplication, determinants, and matrix inverse. Suitable for advanced high school students.

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