To find the determinant of the matrix $B$ given in the image, let's write it out and calculate step-by-step. The matrix $B$ is:

$B = \begin{bmatrix} 2 & 7 & -1 & 4 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 0 \end{bmatrix}$

Since this is a $4 \times 4$ matrix, we’ll need to use cofactor expansion to calculate its determinant.

Steps to calculate:

1. Select a row or column for expansion. Often, it's beneficial to choose the row or column with the most zero entries, but in this case, we can proceed with the first row for simplicity.

2. Cofactor Expansion along the First Row: $\text{det}(B) = 2 \cdot \begin{vmatrix} -9 & 5 & 10 \\ 0 & 1 & -2 \\ -4 & 0 & 0 \end{vmatrix} - 7 \cdot \begin{vmatrix} 3 & 5 & 10 \\ -3 & 1 & -2 \\ 1 & 0 & 0 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 3 & -9 & 10 \\ -3 & 0 & -2 \\ 1 & -4 & 0 \end{vmatrix} + 4 \cdot \begin{vmatrix} 3 & -9 & 5 \\ -3 & 0 & 1 \\ 1 & -4 & 0 \end{vmatrix}$

3. Compute the determinants of each $3 \times 3$ matrix and substitute back to get the final determinant.

I’ll proceed with these calculations.The determinant of the matrix $B$ is approximately $-66$.

Would you like a breakdown of each cofactor expansion step, or do you have any further questions?

Here are some related questions:

1. How does cofactor expansion work for $4 \times 4$ matrices?
2. What is the importance of the determinant in linear algebra?
3. How can we interpret the sign and value of a determinant?
4. What are other methods to calculate the determinant of larger matrices?
5. Why might one prefer using row or column with zeros when expanding?

Tip: For large matrices, use a row or column with the most zeros to simplify the calculation of the determinant.