Answer
$\begin{vmatrix}
4 & -7 & 8\\
2 & 1 & 3\\
-6& 3 & 0
\end{vmatrix}=186$
Work Step by Step
\begin{vmatrix}
4 & -7 & 8\\
2 & 1 & 3\\
-6& 3& 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.
$M_{11}=\begin{vmatrix}
1&3\\3&0\end{vmatrix}=(1)(0)-(3)(3)=0-9=-9$
$M_{21}=\begin{vmatrix}
-7&8\\3&0\end{vmatrix}=(-7)(0)-(8)(3)=0-48=-24$
$M_{31}=\begin{vmatrix}
-7&8\\1&3\end{vmatrix}=(-7)(3)-(8)(1)=-21-8=-29$
Now find the cofactor of each element of these minors.
$A_{11}=(-1)^{1+1} M_{11}=(-1)^2(-9)=-9$
$A_{21}=(-1)^{2+1} M_{21}=(-1)^3(-24)=-(-24)=24$
$A_{31}=(-1)^{3+1} M_{31}=(-1)^4(-29)=-29$
Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.
$\begin{vmatrix}
4 & -7 & 8\\
2 & 1 & 3\\
-6& 3 & 0
\end{vmatrix}= a_{11} A_{11}+a_{21} A_{21}+ a_{31} A_{31}$
$\begin{vmatrix}
4 & -7 & 8\\
2 & 1 & 3\\
-6& 3& 0
\end{vmatrix}=4(-9)+2(24)-6(-29)=-36+48+174=186$
