Answer
$\begin{vmatrix}
8& -2 & -4\\
7 & 0 & 3\\
5& -1& 2
\end{vmatrix}=50$
Work Step by Step
\begin{vmatrix}
8& -2 & -4\\
7 & 0 & 3\\
5& -1& 2
\end{vmatrix}
$\begin{vmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{vmatrix}= a_{11} A_{11}+a_{21} A_{21}+ a_{31} A_{31}$
To find the determinant (expanding by the first column), first find the minor of each element in the first column.
$M_{11}=\begin{vmatrix}
0&3\\-1&2\end{vmatrix}=(0)(2)-(3)(-1)=0+3=3$
$M_{21}=\begin{vmatrix}
-2&-4\\-1&2\end{vmatrix}=(-2)(2)-(-4)(-1)=-4-4=-8$
$M_{31}=\begin{vmatrix}
-2&-4\\0&3\end{vmatrix}=(-2)(3)-(-4)(0)=-6+0=-6$
Now find the cofactor of each element of these minors.
$A_{11}=(-1)^{1+1} M_{11}=(-1)^2(3)=3$
$A_{21}=(-1)^{2+1} M_{21}=(-1)^3(-8)=-(-24)=8$
$A_{31}=(-1)^{3+1} M_{31}=(-1)^4(-6)=-6$
Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.
$\begin{vmatrix}
8& -2 & -4\\
7 & 0 & 3\\
5& -1& 2
\end{vmatrix}= a_{11} A_{11}+a_{21} A_{21}+ a_{31} A_{31}$
$\begin{vmatrix}
8& -2 & -4\\
7 & 0 & 3\\
5& -1& 2
\end{vmatrix}=8(3)+7(8)+5(-6)=24+56-30=50$
