<p>Evaluate the determinant:</p><p>\(\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix}\)</p>
Step-by-Step Solution
Key Concept: Apply cofactor expansion along any row or column; multiply each element by its cofactor (minor with appropriate sign).
<p><strong>Expanding along the first row:</strong></p><p>$\Delta = 1 \begin{vmatrix} 5 & 6 \\ 8 & 0 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6 \\ 7 & 0 \end{vmatrix} + 3 \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix}$</p><p>$= 1(0 - 48) - 2(0 - 42) + 3(32 - 35)$</p><p>$= -48 + 84 - 9$</p><p>$= 27$</p><p>∴ The determinant is $27 \neq 0$, so Statement-1 is true.</p>
Correct Answer: 27