The value of $\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix}$ is
Step-by-Step Solution
Key Concept: To find the determinant of a 3×3 matrix, expand along any row or column using the cofactor method. We'll expand along the first row, calculating 2×2 minors and applying alternating signs.
<p><strong>Step 1:</strong> Set up the determinant expansion along the first row:</p><p>$$\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix} = 1\begin{vmatrix} 3 & 6 \\ -7 & 9 \end{vmatrix} - 2\begin{vmatrix} -4 & 6 \\ 2 & 9 \end{vmatrix} + 3\begin{vmatrix} -4 & 3 \\ 2 & -7 \end{vmatrix}$$</p><p><strong>Step 2:</strong> Calculate the first 2×2 minor:</p><p>$$\begin{vmatrix} 3 & 6 \\ -7 & 9 \end{vmatrix} = (3)(9) - (6)(-7) = 27 + 42 = 69$$</p><p><strong>Step 3:</strong> Calculate the second 2×2 minor:</p><p>$$\begin{vmatrix} -4 & 6 \\ 2 & 9 \end{vmatrix} = (-4)(9) - (6)(2) = -36 - 12 = -48$$</p><p><strong>Step 4:</strong> Calculate the third 2×2 minor:</p><p>$$\begin{vmatrix} -4 & 3 \\ 2 & -7 \end{vmatrix} = (-4)(-7) - (3)(2) = 28 - 6 = 22$$</p><p><strong>Step 5:</strong> Substitute and compute the full determinant:</p><p>$$= 1(69) - 2(-48) + 3(22)$$</p><p>$$= 69 + 96 + 66$$</p><p>$$= 231$$</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C