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We are asked to find the determinant of the matrix expression:

\[
\left[(ABC)^{-1}(4A)\right]^2(C^T),
\]
where \( \det(A) = -1 \), \( \det(B) = 4 \), and \( \det(C) = 8 \).

### Step-by-step breakdown:

#### 1. Determinant of \( ABC \):
Using the property of determinants for matrix products:
\[
\det(ABC) = \det(A) \cdot \det(B) \cdot \det(C).
\]
Substitute the given values:
\[
\det(ABC) = (-1) \cdot 4 \cdot 8 = -32.
\]

#### 2. Determinant of \( (ABC)^{-1} \):
The determinant of the inverse of a matrix is the reciprocal of the determinant:
\[
\det\left((ABC)^{-1}\right) = \frac{1}{\det(ABC)} = \frac{1}{-32} = -\frac{1}{32}.
\]

#### 3. Determinant of \( 4A \):
The determinant of a scalar multiplied by a matrix is given by:
\[
\det(4A) = 4^3 \cdot \det(A) = 64 \cdot (-1) = -64.
\]
(Note: The power 3 comes from the matrix being \( 3 \times 3 \).)

#### 4. Determinant of \( (ABC)^{-1}(4A) \):
Using the property of determinants for matrix products:
\[
\det\left((ABC)^{-1}(4A)\right) = \det\left((ABC)^{-1}\right) \cdot \det(4A).
\]
Substitute the values:
\[
\det\left((ABC)^{-1}(4A)\right) = \left(-\frac{1}{32}\right) \cdot (-64) = 2.
\]

#### 5. Determinant of \( \left[(ABC)^{-1}(4A)\right]^2 \):
The determinant of a matrix raised to a power is the determinant raised to that power:
\[
\det\left(\left[(ABC)^{-1}(4A)\right]^2\right) = \left(\det\left((ABC)^{-1}(4A)\right)\right)^2 = 2^2 = 4.
\]

#### 6. Determinant of \( C^T \):
The determinant of a transpose of a matrix is the same as the determinant of the matrix:
\[
\det(C^T) = \det(C) = 8.
\]

#### 7. Final step: Determinant of the entire expression:
Using the property of determinants for matrix products again:
\[
\det\left(\left[(ABC)^{-1}(4A)\right]^2(C^T)\right) = \det\left(\left[(ABC)^{-1}(4A)\right]^2\right) \cdot \det(C^T).
\]
Substitute the values:
\[
\det\left(\left[(ABC)^{-1}(4A)\right]^2(C^T)\right) = 4 \cdot 8 = 32.
\]

### Final Answer:
\[
\boxed{32}.
\]

Let me know if you'd like any details or have further questions.

Here are 5 related questions to expand your understanding:
1. How does the determinant change when a matrix is multiplied by a scalar?
2. What is the relationship between the determinant of a matrix and its inverse?
3. How do we compute the determinant of a product of matrices?
4. What properties of transposes help in determinant calculations?
5. Can you explain why determinants are multiplicative over matrix products?

**Tip**: When raising a matrix to a power, remember that the determinant of the matrix is also raised to that power.

Suppose A, B and C are 3 × 3 matrices such that det A = -1, det B = 4 and det C = 8. Find the determinant of the matrix [(ABC)^{-1}(4A)]^2 (C^T).

Solve a problem involving the determinants of 3x3 matrices A, B, and C. The solution explores determinant properties including inverses, transposes, and scalar multiplications. Ideal for university-level linear algebra studies.

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