find the determinant of the matrix [0 4 -1 5;-4 0 3 -2; 1 -3 0 1; -5 2 -1 0]

The matrix provided is a 4x4 matrix:

\[
A = \begin{bmatrix}
0 & 4 & -1 & 5 \\
-4 & 0 & 3 & -2 \\
1 & -3 & 0 & 1 \\
-5 & 2 & -1 & 0
\end{bmatrix}
\]

We can find the determinant of a 4x4 matrix using cofactor expansion. I'll use the first row for expansion:

\[
\text{det}(A) = 0 \cdot \text{det}(M_{11}) - 4 \cdot \text{det}(M_{12}) + (-1) \cdot \text{det}(M_{13}) + 5 \cdot \text{det}(M_{14})
\]

where \( M_{ij} \) represents the 3x3 minor matrix obtained by removing the \(i\)-th row and \(j\)-th column from the original matrix.

Since the first term involves \( 0 \), we only need to calculate the determinants of the 3x3 minors \( M_{12} \), \( M_{13} \), and \( M_{14} \).

Let me calculate this step-by-step.The determinant of the matrix is approximately \( \text{det}(A) = 289 \).

Would you like further details on how this determinant was computed?

Here are some related questions to deepen your understanding:

1. How is cofactor expansion applied to a 4x4 matrix in more detail?
2. How do properties of matrices affect the determinant?
3. What happens if a matrix has repeated rows or columns in terms of its determinant?
4. Can determinants be used to check if a matrix is invertible?
5. How can row operations simplify the calculation of determinants?

**Tip:** If a matrix has a row or column of zeros, the determinant is zero, which can simplify calculations.

find the determinant of the matrix [0 4 -1 5; -4 0 3 -2; 1 -3 0 1; -5 2 -1 0]

Discover how to find the determinant of the 4x4 matrix [0 4 -1 5; -4 0 3 -2; 1 -3 0 1; -5 2 -1 0] using cofactor expansion. This problem covers essential linear algebra concepts such as matrices and determinants, and is suitable for high school and undergraduate students.

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