<p>If \(\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\), then</p>
Step-by-Step Solution
Key Concept: When evaluating determinants with complex entries, carefully apply the expansion formula (Sarrus rule or cofactor) without mixing up imaginary unit operations. The determinant itself becomes a complex number that must be simplified to the form x + iy.
<p><strong>Step 1:</strong> Expand the determinant along the first row using cofactor expansion:</p><p>det = 6i·|3i, -1; 3, i| - (-3i)·|4, -1; 20, i| + 1·|4, 3i; 20, 3|</p><p><strong>Step 2:</strong> Calculate each 2×2 determinant:</p><p>• First: (3i)(i) - (-1)(3) = 3i² + 3 = -3 + 3 = 0</p><p>• Second: (4)(i) - (-1)(20) = 4i + 20</p><p>• Third: (4)(3) - (3i)(20) = 12 - 60i</p><p><strong>Step 3:</strong> Substitute back:</p><p>det = 6i(0) + 3i(4i + 20) + 1(12 - 60i)</p><p>= 0 + 12i² + 60i + 12 - 60i</p><p>= -12 + 60i + 12 - 60i</p><p>= 0</p><p><strong>Step 4:</strong> Therefore x = 0 and y = 0, so x + iy = 0</p><p>∴ Answer: D</p>
Correct Answer: D