Matrices & Determinants
Determinant Calculation
Grade 12
Question:
<p><strong>Question 87:</strong> <strong>Statement-1:</strong> The value of the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix} = 2$.</p><p><strong>Statement-2:</strong> Neither of two rows or columns of [a matrix property regarding determinants].</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: Direct calculation of a 3×3 determinant using cofactor expansion.
<p>Compute $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix}$ using expansion along row 3:</p><p>$\Delta = 7(2 \cdot 6 - 3 \cdot 5) - 8(1 \cdot 6 - 3 \cdot 4) + 0 = 7(12-15) - 8(6-12) = 7(-3) - 8(-6) = -21 + 48 = 27$</p><p>So Statement-1 is false (the determinant is 27, not 2).</p><p>Statement-2 is incomplete or unclear in the context provided.</p><p>The answer is (c) or (d) depending on Statement-2 verification.</p>
Correct Answer: c