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Let's work through this problem step by step. You need to find the determinants of matrices \( A \), \( B \), and \( AB \), and verify that \( |A||B| = |AB| \).

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

### Steps:

1. **Find the determinant of matrix \( A \), \( |A| \)**.
2. **Find the determinant of matrix \( B \), \( |B| \)**.
3. **Multiply matrices \( A \) and \( B \) to get matrix \( AB \)** (which you have calculated).
4. **Find the determinant of \( AB \), \( |AB| \)**.
5. **Check if \( |A| \times |B| = |AB| \)**.

I'll begin by calculating the determinants of \( A \) and \( B \), and then verify the relationship. Let me calculate this for you.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like me to guide you through manually calculating the determinants and verifying the relationship? Let me know if you have any other questions as well!

Here are five related questions for further practice:
1. How do you calculate the determinant of a 4x4 matrix?
2. What is the geometric interpretation of the determinant of a matrix?
3. What happens to the determinant of a product of two matrices?
4. How would the determinant change if we transposed one of the matrices?
5. Can you determine if a matrix is invertible based on its determinant?

**Tip:** When multiplying two matrices, remember that the determinant of the product equals the product of the determinants only if both matrices are square.

Find |A|, |B|, AB, and |AB|. Then verify that |A||B| = |AB| using the matrices A and B provided.

Learn how to calculate the determinants of matrices A and B, find their product AB, and verify that |A||B| = |AB|. This problem involves matrix algebra and the properties of determinants, suitable for college-level linear algebra students.

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