Answer
$\begin{vmatrix}
1 & 2 & 0\\
-1 & 2 & -1\\
0& 1& 4
\end{vmatrix}=17$
Work Step by Step
\begin{vmatrix}
1 & 2 & 0\\
-1 & 2 & -1\\
0& 1& 4
\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.
$M_{11}=\begin{vmatrix}
2&-1\\1&4\end{vmatrix}=(2)(4)-(-1)(1)=8+1=9$
$M_{21}=\begin{vmatrix}
2&0\\1&4\end{vmatrix}=(2)(4)-(0)(1)=8-0=8$
$M_{31}=\begin{vmatrix}
2&0\\2&-1\end{vmatrix}=(2)(-1)-(0)(2)=-2-0=-2$
Now find the cofactor of each element of these minors.
$A_{11}=(-1)^{1+1} M_{11}=(-1)^2(9)=9$
$A_{21}=(-1)^{2+1} M_{21}=(-1)^3(8)=-(8)=-8$
$A_{31}=(-1)^{3+1} M_{31}=(-1)^4(-2)=-2$
Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.
$\begin{vmatrix}
1 & 2 & 0\\
-1 & 2 & -1\\
0& 1& 4
\end{vmatrix}= a_{11} A_{11}+a_{21} A_{21}+ a_{31} A_{31}$
$\begin{vmatrix}
1 & 2 & 0\\
-1 & 2 & -1\\
0& 1& 4
\end{vmatrix}=1(9)-1(-8)+0(-2)=9+8+0=17$
