Answer
$\begin{vmatrix}
2& 1 & -1\\
4 & 7 & -2\\
2& 4& 0
\end{vmatrix}=10$
Work Step by Step
\begin{vmatrix}
2& 1 & -1\\
4 & 7 & -2\\
2& 4& 0
\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.
$M_{11}=\begin{vmatrix}
7&-2\\4&0\end{vmatrix}=(7)(0)-(-2)(4)=0-(-8)=8$
$M_{21}=\begin{vmatrix}
1&-1\\4&0\end{vmatrix}=(1)(0)-(-1)(4)=0-(-4)=4$
$M_{31}=\begin{vmatrix}
1&-1\\7&-2\end{vmatrix}=(1)(-2)-(-1)(7)=-2-(-7)=-2+7=5$
Now find the cofactor of each element of these minors.
$A_{11}=(-1)^{1+1} M_{11}=(-1)^2(8)=8$
$A_{21}=(-1)^{2+1} M_{21}=(-1)^3(4)=-(4)=-4$
$A_{31}=(-1)^{3+1} M_{31}=(-1)^4(5)=5$
Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.
$\begin{vmatrix}
2& 1 & -1\\
4 & 7 & -2\\
2& 4& 0
\end{vmatrix}= a_{11} A_{11}+a_{21} A_{21}+ a_{31} A_{31}$
$\begin{vmatrix}
2& 1 & -1\\
4 & 7 & -2\\
2& 4& 0
\end{vmatrix}=2(8)+4(-4)+2(10)=16-16+10=10$
