Matrices & Determinants
Determinants with complex numbers
Grade 12
Question:
<p>If <br>\[\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\] then</p>
<p>(a) \(x = 3, y = 1\)</p>
<p>(b) \(x = 1, y = 3\)</p>
<p>(c) \(x = 0, y = 3\)</p>
<p>(d) \(x = 0, y = 0\)</p>
Step-by-Step Solution
Key Concept: Expand the determinant using the standard cofactor method along any row/column, then carefully separate real and imaginary parts from complex arithmetic involving powers of i (where i² = -1, i³ = -i, i⁴ = 1).
<p><strong>Step 1:</strong> Expand along the first row:</p><p>Δ = 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>|3i -1; 3 i| = (3i)(i) - (-1)(3) = 3i² + 3 = -3 + 3 = 0</p><p>|4 -1; 20 i| = (4)(i) - (-1)(20) = 4i + 20 = 20 + 4i</p><p>|4 3i; 20 3| = (4)(3) - (3i)(20) = 12 - 60i</p><p><strong>Step 3:</strong> Substitute back:</p><p>Δ = 6i(0) + 3i(20 + 4i) + 1(12 - 60i)</p><p>Δ = 0 + 60i + 12i² + 12 - 60i</p><p>Δ = 60i - 12 + 12 - 60i = -12</p><p>∴ x = -12, y = 0, so the answer is <strong>D</strong></p>
Correct Answer: D