Answer
The easiest way to evaluate
$\left| \begin{matrix}
3 & 2 & 8 \\
5 & -4 & 0 \\
-6 & 7 & 0 \\
\end{matrix} \right|$
is to expand about the elements in column 3.
Work Step by Step
To calculate the value for a given determinant, choose any row or column and calculate the co-factors. The co-factor of a determinant $A$ is:
$A=\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
{{b}_{21}} & {{b}_{22}} & {{b}_{23}} \\
{{c}_{31}} & {{c}_{32}} & {{c}_{33}} \\
\end{matrix} \right|$
$C\left( {{a}_{11}} \right)={{\left( -1 \right)}^{i+j}}\left| \begin{matrix}
{{b}_{22}} & {{b}_{23}} \\
{{c}_{32}} & {{c}_{33}} \\
\end{matrix} \right|$, where $i,j$ are row and column.
Now multiply the co-factors with the same number:
$A={{a}_{11}}C\left( {{a}_{11}} \right)+{{a}_{12}}C\left( {{a}_{12}} \right)+{{a}_{13}}C\left( {{a}_{13}} \right)$
Therefore, column 3 consists of two zeroes, so we need to calculate only one co-factor. Thus, it is easier to calculate.
