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We are tasked with solving the expression:

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

We are provided with the inverses of the matrices \( A^{-1} \), \( B^{-1} \), and \( C^{-1} \), but we do not need to find \( A \), \( B \), or \( C \) explicitly. Instead, we'll use matrix properties and inverses to simplify the expression.

### Step-by-Step Approach:

1. **Rewrite the Expression:**

   \[
   \left( \left( \frac{1}{2} A \right)^2 C (2B)^T \right)^{-1}
   \]
   
   Using matrix properties:
   - \( \left( \frac{1}{2} A \right)^2 = \frac{1}{4} A^2 \)
   - \( (2B)^T = 2B^T \)
   
   This simplifies the expression to:

   \[
   \left( \frac{1}{4} A^2 C 2 B^T \right)^{-1}
   = \left( \frac{1}{2} A^2 C B^T \right)^{-1}
   \]

2. **Apply the Inverse:**

   The inverse of a product of matrices \( (XYZ)^{-1} = Z^{-1} Y^{-1} X^{-1} \), so applying the inverse:

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

3. **Simplify Further:**
   - \( (B^T)^{-1} = (B^{-1})^T \) (since the inverse of a transpose is the transpose of the inverse),
   - \( (A^2)^{-1} = (A^{-1})^2 \) (since the inverse of \( A^2 \) is the square of \( A^{-1} \)).
   
   Thus, the expression becomes:

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

### Conclusion:

The final simplified expression is:

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

You can now substitute the given values of \( A^{-1} \), \( B^{-1} \), and \( C^{-1} \) to compute the final result.

Would you like the explicit calculation with the matrix values? Let me know if you'd like more details on any step!

### Related Questions:
1. What is the inverse of a product of two matrices?
2. How do you compute the transpose of a matrix product?
3. Why does \( (A^2)^{-1} = (A^{-1})^2 \) hold?
4. How can the inverse of a scalar multiple of a matrix be computed?
5. How do matrix inverses interact with transposes?

### Tip:
Always remember that the inverse of a product of matrices is the product of the inverses in reverse order!

Suppose A, B, C are 3x3 matrices such that A^-1, B^-1, and C^-1 are given. Then solve ((1/2 A)^2 C(2B)^T)^-1.

This problem involves solving a complex matrix inverse expression using properties of matrix inverses, transposes, and scalar multiplication. Key steps include simplifying the matrix expression and applying the inverse of a product of matrices. Suitable for advanced undergraduate-level students.

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