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The problem asks to solve the expression:

\[
\left( \left( \frac{1}{2} A \right)^2 C (2B)^T \right)^{-1}
\]

Given the matrices \( A^{-1}, B^{-1}, C^{-1} \), we first need to derive \( A, B, C \) by inverting these matrices, then compute the expression step by step.

### Matrices given:

1. \( A^{-1} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ -1 & -1 & 1 \end{bmatrix} \)
2. \( B^{-1} = \begin{bmatrix} 1 & 0 & -1 \\ -2 & -1 & 0 \\ -2 & -1 & -1 \end{bmatrix} \)
3. \( C^{-1} = \begin{bmatrix} 1 & 1 & 0 \\ -2 & 2 & -2 \\ 0 & 1 & -2 \end{bmatrix} \)

### Steps:

1. **Find \( A \), \( B \), and \( C \)**:
   - The inverse of \( A^{-1} \), \( B^{-1} \), and \( C^{-1} \) will give us \( A \), \( B \), and \( C \), respectively.
  
2. **Compute \( \frac{1}{2} A \)**:
   - After obtaining \( A \), multiply it by \( \frac{1}{2} \).

3. **Square \( \left( \frac{1}{2} A \right)^2 \)**:
   - Matrix multiplication of \( \frac{1}{2} A \) with itself.

4. **Find the transpose of \( 2B \)**:
   - After obtaining \( B \), multiply by 2 and find the transpose.

5. **Multiply \( \left( \frac{1}{2} A \right)^2 \), \( C \), and \( (2B)^T \)**:
   - Multiply these matrices in the order provided.

6. **Take the inverse of the final result**.

Let me perform these steps now.The solution to the matrix expression \(\left( \left( \frac{1}{2} A \right)^2 C (2B)^T \right)^{-1}\) is:

\[
\begin{bmatrix}
-32 & -16 & -26 \\
-14 & -6 & -16 \\
-2  & 2  & -14
\end{bmatrix}
\]

If you need further details or have questions, feel free to ask!

Here are five related questions that could extend your understanding:

1. How do you compute the inverse of a 3x3 matrix manually?
2. What properties of matrix inverses and transposes are essential in matrix algebra?
3. How does matrix multiplication differ from scalar multiplication?
4. Why is the determinant of a matrix important in determining whether a matrix is invertible?
5. What role does the transpose of a matrix play in solving matrix expressions?

**Tip:** When performing matrix operations involving inverses, always check if the determinant is non-zero to ensure the matrix is invertible.

What is the answer to the matrix expression \left( \left( \frac{1}{2} A \right)^2 C (2B)^T \right)^{-1}?

This problem asks to solve the matrix expression \left( \left( \frac{1}{2} A \right)^2 C (2B)^T \right)^{-1}. The solution involves key linear algebra concepts such as matrix inversion, transposition, scalar multiplication, and matrix multiplication.

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