Given that $A$ and $B$ are square matrices of order 4 with determinants $|A| = -6$ and $|B| = -6$, we can use the properties of determinants to find the required values.

(a) $|AB|$

The determinant of the product of two matrices is the product of their determinants:

$|AB| = |A| \cdot |B| = (-6) \cdot (-6) = 36$

(b) $|A^3|$

The determinant of a matrix raised to a power $n$ is the determinant of the matrix raised to that power:

$|A^3| = |A|^3 = (-6)^3 = -216$

(c) $|3B|$

If $c$ is a scalar and $B$ is a matrix of order $n$, then the determinant of the scalar multiple is given by:

$|cB| = c^n \cdot |B|$

Since $B$ is a 4x4 matrix, $n = 4$, and $c = 3$:

$|3B| = 3^4 \cdot |B| = 81 \cdot (-6) = -486$

(d) $|(AB)^T|$

The determinant of a matrix is equal to the determinant of its transpose:

$|(AB)^T| = |AB| = 36$

(e) $|A^{-1}|$

The determinant of the inverse of a matrix is the reciprocal of the determinant:

$|A^{-1}| = \frac{1}{|A|} = \frac{1}{-6} = -\frac{1}{6}$

Summary of Results:

$\text{(a)}\ |AB| = 36$ $\text{(b)}\ |A^3| = -216$ $\text{(c)}\ |3B| = -486$ $\text{(d)}\ |(AB)^T| = 36$ $\text{(e)}\ |A^{-1}| = -\frac{1}{6}$

Would you like further explanation on any of these points, or do you have any other questions?

Here are 8 follow-up questions related to this topic:

1. How does the determinant of a matrix change if the matrix is multiplied by a scalar?
2. What does the determinant tell us about the invertibility of a matrix?
3. How would you find the determinant of a product of three matrices?
4. What happens to the determinant if a matrix is transposed?
5. What is the determinant of the identity matrix of any order?
6. How do you compute the determinant of a block matrix?
7. Can a matrix with a determinant of zero be inverted? Why or why not?
8. What are the implications of a negative determinant for a matrix?

Tip: Remember that the determinant provides important insights into the matrix, such as its invertibility and how it scales volume in linear transformations.