To solve this problem, we need to calculate the determinants of the matrices provided in the image.

Matrix $A$

$A = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 3 & 2 & 1 \\ 1 & 4 & 0 & 2 \\ 2 & 0 & 0 & 4 \end{pmatrix}$

The determinant of a $4 \times 4$ matrix can be computed by expanding along a row or a column. Given the zeros in the first row, it’s advantageous to expand along that row.

Step-by-Step Solution

1. Expand along the first row (using cofactor expansion): $\det(A) = 0 \cdot M_{11} - 0 \cdot M_{12} + 0 \cdot M_{13} + 1 \cdot M_{14}$ where $M_{ij}$ represents the minor of element $a_{ij}$, which is the determinant of the $3 \times 3$ matrix obtained by removing the $i$-th row and $j$-th column.

   Since only the term $1 \cdot M_{14}$ remains, we need only calculate the minor $M_{14}$.

2. Calculate $M_{14}$: Removing the first row and fourth column, the $3 \times 3$ submatrix is: $\begin{pmatrix} 0 & 3 & 2 \\ 1 & 4 & 0 \\ 2 & 0 & 0 \end{pmatrix}$

3. Find the determinant of this $3 \times 3$ matrix: $\det\begin{pmatrix} 0 & 3 & 2 \\ 1 & 4 & 0 \\ 2 & 0 & 0 \end{pmatrix} = 0 \cdot \begin{vmatrix} 4 & 0 \\ 0 & 0 \end{vmatrix} - 3 \cdot \begin{vmatrix} 1 & 0 \\ 2 & 0 \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & 4 \\ 2 & 0 \end{vmatrix}$

4. Calculate each minor:

   • For $0 \cdot \begin{vmatrix} 4 & 0 \\ 0 & 0 \end{vmatrix} = 0$
   • For $-3 \cdot \begin{vmatrix} 1 & 0 \\ 2 & 0 \end{vmatrix} = -3 \cdot (1 \cdot 0 - 0 \cdot 2) = 0$
   • For $2 \cdot \begin{vmatrix} 1 & 4 \\ 2 & 0 \end{vmatrix} = 2 \cdot (1 \cdot 0 - 4 \cdot 2) = 2 \cdot (-8) = -16$

   Therefore, $\det\begin{pmatrix} 0 & 3 & 2 \\ 1 & 4 & 0 \\ 2 & 0 & 0 \end{pmatrix} = -16$

5. Complete the determinant calculation for $A$: $\det(A) = 1 \cdot (-16) = -16$

Conclusion

The determinant of matrix $A$ is $\det(A) = -16$.

Would you like to proceed with the determinant calculation for matrix $B$ as well?

Here are five related questions to deepen understanding:

1. How does cofactor expansion work for determinants of larger matrices?
2. What properties of determinants can simplify calculations?
3. How does the presence of zero elements affect cofactor expansion?
4. Can determinants be used to solve systems of linear equations?
5. What does the determinant tell us about the invertibility of a matrix?

Tip: Expanding along rows or columns with the most zeros simplifies determinant calculations.