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} )

Question

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} )

Accepted Answer

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 )

Answer

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 )

Find the determinant of the matrix $A = [\begin{matrix} 3 & 0 & - 1 & 4 1 & 5 & 0 & - 2 4 & 1 & 2 & 1 2 & 0 & - 1 & 3 \end{matrix}]$

Answer

Popular Problems

Find the determinant of the matrix A=[30−14 150−2 4121 20−13]

Answer

Popular Problems

Find the determinant of the matrix $A = [\begin{matrix} 3 & 0 & - 1 & 4 1 & 5 & 0 & - 2 4 & 1 & 2 & 1 2 & 0 & - 1 & 3 \end{matrix}]$