Find the determinant of the matrix \( A = \begin{bmatrix} 3 & 0 & -1 & 4 \ 1 & 5 & 0 & -2 \ 4 & 1 & 2 & 1 \ 2 & 0 & -1 & 3 \end{bmatrix} \)

Step 1 ：Expand the determinant using the first row: \( |A| = 3|A_{11}| - 0|A_{12}| + (-1)|A_{13}| + 4|A_{14}| \)

Step 2 ：Compute the sub-matrices: \( A_{11} = \begin{bmatrix} 5 & 0 & -2 \ 1 & 2 & 1 \ 0 & -1 & 3 \end{bmatrix}, \ A_{13} = \begin{bmatrix} 1 & 5 & -2 \ 4 & 1 & 1 \ 2 & 0 & 3 \end{bmatrix}, \ A_{14} = \begin{bmatrix} 1 & 5 & 0 \ 4 & 1 & 2 \ 2 & 0 & -1 \end{bmatrix} \)

Step 3 ：Compute the determinants of the sub-matrices: \( |A_{11}| = 5(3+2) - 0 + (-2)(-2-0) = 39, \ |A_{13}| = 1(3+0) - 5(-2-6) +(-2)(8-2) = 37, \ |A_{14}| = 1(3+0) - 5(-1-4) + 0 = 26 \)

Step 4 ：Substitute the determinants of the sub-matrices into the expansion: \( |A| = 3*39 - 0 + (-1)*37 + 4*26 = 117 -37 + 104 = 184 \)