The matrix provided in the image is:

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

To find the determinant of this 3x3 matrix $B$, we can use the cofactor expansion along the first row. The determinant of a 3x3 matrix is given by:

$\text{det}(B) = (-1) \cdot \text{det} \begin{pmatrix} 1 & -2 \\ 5 & 10 \end{pmatrix} - 0 \cdot \text{det} \begin{pmatrix} 3 & -2 \\ -9 & 10 \end{pmatrix} + 2 \cdot \text{det} \begin{pmatrix} 3 & 1 \\ -9 & 5 \end{pmatrix}$

Step 1: Calculate each 2x2 determinant

1. $\text{det} \begin{pmatrix} 1 & -2 \\ 5 & 10 \end{pmatrix} = (1)(10) - (-2)(5) = 10 + 10 = 20$

2. The second term will be zero, because the cofactor's multiplier is zero.

3. $\text{det} \begin{pmatrix} 3 & 1 \\ -9 & 5 \end{pmatrix} = (3)(5) - (1)(-9) = 15 + 9 = 24$

Step 2: Substitute into the cofactor expansion

$\text{det}(B) = (-1)(20) + 0 + 2(24) = -20 + 48 = 28$

Thus, the determinant of matrix $B$ is $\boxed{28}$.

Would you like a more detailed explanation or have any other questions?

Here are five related questions you might find interesting:

1. How do we apply cofactor expansion for larger matrices (4x4 or more)?
2. What are some properties of determinants that simplify calculations?
3. How is the determinant useful in solving systems of linear equations?
4. How does the determinant change when you swap two rows of a matrix?
5. What is the geometrical interpretation of the determinant?

Tip: The determinant of a matrix tells us about the volume scaling factor and whether the transformation is orientation-preserving (positive determinant) or reversing (negative determinant).