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.