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To find the determinant of the given 4x4 matrix, we will use cofactor expansion. The matrix provided is:

\[
\begin{pmatrix}
0 & 3 & -1 & 4 \\
-3 & 2 & 1 & 5 \\
0 & 3 & -7 & 1 \\
2 & 8 & 9 & 0
\end{pmatrix}
\]

We'll expand along the first row (since it contains a zero, which simplifies the calculation):

\[
\text{det}(A) = 0 \cdot \text{det}(M_{11}) - 3 \cdot \text{det}(M_{12}) + (-1) \cdot \text{det}(M_{13}) + 4 \cdot \text{det}(M_{14})
\]

Since the first term is multiplied by 0, it can be ignored. Let's compute the determinants of the 3x3 minors \( M_{12} \), \( M_{13} \), and \( M_{14} \).

### Minor \( M_{12} \):
Removing the first row and second column, the minor \( M_{12} \) is:

\[
\begin{pmatrix}
-3 & 1 & 5 \\
0 & -7 & 1 \\
2 & 9 & 0
\end{pmatrix}
\]

\[
\text{det}(M_{12}) = -3 \left((-7)(0) - (1)(9)\right) - 1 \left(0 \cdot 0 - 1 \cdot 2\right) + 5 \left(0 \cdot 9 - (-7) \cdot 2\right)
\]
\[
\text{det}(M_{12}) = -3 \cdot (-9) - 1 \cdot (-2) + 5 \cdot 14
\]
\[
\text{det}(M_{12}) = 27 + 2 + 70 = 99
\]

### Minor \( M_{13} \):
Removing the first row and third column, the minor \( M_{13} \) is:

\[
\begin{pmatrix}
-3 & 2 & 5 \\
0 & 3 & 1 \\
2 & 8 & 0
\end{pmatrix}
\]

\[
\text{det}(M_{13}) = -3 \left(3 \cdot 0 - 1 \cdot 8\right) - 2 \left(0 \cdot 0 - 1 \cdot 2\right) + 5 \left(0 \cdot 8 - 3 \cdot 2\right)
\]
\[
\text{det}(M_{13}) = -3 \cdot (-8) - 2 \cdot (-2) + 5 \cdot (-6)
\]
\[
\text{det}(M_{13}) = 24 + 4 - 30 = -2
\]

### Minor \( M_{14} \):
Removing the first row and fourth column, the minor \( M_{14} \) is:

\[
\begin{pmatrix}
-3 & 2 & 1 \\
0 & 3 & -7 \\
2 & 8 & 9
\end{pmatrix}
\]

\[
\text{det}(M_{14}) = -3 \left(3 \cdot 9 - (-7) \cdot 8\right) - 2 \left(0 \cdot 9 - (-7) \cdot 2\right) + 1 \left(0 \cdot 8 - 3 \cdot 2\right)
\]
\[
\text{det}(M_{14}) = -3 \cdot (27 + 56) - 2 \cdot (14) - 6
\]
\[
\text{det}(M_{14}) = -3 \cdot 83 - 28 - 6 = -249 - 34 = -283
\]

### Putting it all together:

\[
\text{det}(A) = -3 \cdot 99 + (-1) \cdot (-2) + 4 \cdot (-283)
\]
\[
\text{det}(A) = -297 + 2 - 1132 = -1427
\]

Thus, the determinant of the matrix is **-1427**.

Would you like more details or have any questions about the steps? 

Here are five related questions to further explore the topic:
1. How does expanding along different rows or columns affect the calculation of the determinant?
2. What properties of determinants can be useful for simplifying calculations?
3. How does the determinant relate to the invertibility of a matrix?
4. Can the determinant help in solving systems of linear equations?
5. What are some practical applications of determinants in physics and engineering?

**Tip:** Always look for rows or columns with zeros when calculating determinants, as they can simplify the expansion process.

Learn how to find the determinant of a 4x4 matrix using cofactor expansion with a step-by-step solution. This problem covers matrix determinants, cofactor expansion, and advanced algebraic concepts. Suitable for advanced high school students and beyond.

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