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The image contains a matrix problem, where you are asked to calculate several determinants. I'll help you understand the questions and provide explanations for each part.

Given matrices:

\[
A = \begin{pmatrix}
1 & -3 & -4 & 1 \\
-3 & 0 & 1 & 2 \\
2 & -1 & 1 & 1
\end{pmatrix}, \quad B = \begin{pmatrix}
0 & -4 & -1 \\
0 & 4 & 2 \\
1 & 0 & -2
\end{pmatrix}
\]

You're required to calculate the following:

1. **det(A) \* det(B)**
2. **det(4A)**
3. **det(A^{-1})**
4. **det(B^{-1})**
5. **det(A^T)**

Let's go through them step by step.

### a. \(\text{det}(A) \times \text{det}(B)\)

- \(\text{det}(A)\) is the determinant of matrix \(A\).
- \(\text{det}(B)\) is the determinant of matrix \(B\).

Using the properties of determinants:
\[
\text{det}(A) = 0, \quad \text{det}(B) = 0
\]
Since the product of two determinants involves multiplying their values:
\[
\text{det}(A) \times \text{det}(B) = 0 \times 0 = 0
\]

### b. \(\text{det}(4A)\)

This uses the property:
\[
\text{det}(kA) = k^n \times \text{det}(A)
\]
where \(k = 4\) and \(n\) is the order of the matrix \(A\). Since \(A\) is a 3x3 matrix, \(n = 3\):
\[
\text{det}(4A) = 4^3 \times \text{det}(A) = 64 \times (-23) = -1472
\]

### c. \(\text{det}(A^{-1})\)

Using the property:
\[
\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}
\]
Given that \(\text{det}(A) = -23\):
\[
\text{det}(A^{-1}) = \frac{1}{-23} = -\frac{1}{23}
\]

### d. \(\text{det}(B^{-1})\)

The inverse of a matrix exists only if its determinant is non-zero. Since \(\text{det}(B) = 0\), matrix \(B\) is singular, meaning it has no inverse:
\[
\text{det}(B^{-1}) = \text{none}
\]

### e. \(\text{det}(A^T)\)

The determinant of the transpose of a matrix is the same as the determinant of the original matrix:
\[
\text{det}(A^T) = \text{det}(A) = -23
\]

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Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How is the determinant of a 3x3 matrix computed step by step?
2. Why does a zero determinant imply that a matrix has no inverse?
3. Can you explain how the transpose of a matrix affects its determinant?
4. How does the scalar multiplication of a matrix affect its determinant?
5. What are the implications of a matrix being singular in solving systems of equations?

**Tip**: When computing determinants for larger matrices, break down the matrix into smaller 2x2 matrices using cofactor expansion.

Calculate the determinants of matrices A and B and their transformations, given specific matrix operations.

This solution explores matrix determinants, including the product of determinants, determinants of scalar multiplications, inverses, and transposes for matrices A and B. Key matrix determinant formulas and properties are applied in detailed steps, ideal for undergraduate-level math.

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