Let *A* and *B* be square matrices of order 4 such that |*A*| = –6 and |*B*| = –6. Find the following.

(a)    |*AB*|

(b)    |*A*3|

(c)    |3*B*|

(d)    |(*AB*)T|

(e)    |*A*−1|

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.

Explore how to calculate determinants of matrices, including products, powers, scalar multiples, transposes, and inverses. Understand the properties and applications of determinants with detailed examples and explanations.

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