Evaluate the determinant $\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix}$.
Step-by-Step Solution
Key Concept: General
$\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix} = 1 \begin{vmatrix} 3 & 6 \\ -7 & 9 \end{vmatrix} - 2 \begin{vmatrix} -4 & 6 \\ 2 & 9 \end{vmatrix} + 3 \begin{vmatrix} -4 & 3 \\ 2 & -7 \end{vmatrix} = (27 + 42) - 2(-36 - 12) + 3(28 - 6) = 231$.<br/>Alternative: By Sarrus diagram, the value is $(27 + 24 + 84) - (18 - 42 - 72) = 135 - (18 - 114) = 231$.
Correct Answer: 231