Matrices & Determinants
Determinants with Complex Entries
Grade 12
Question:
<p>If <span style='display:inline-block; border: 1px solid black; padding: 5px;'>\[\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\]</span> where \(i = \sqrt{-1}\), 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 a complex determinant carefully, using $i^2 = -1$ and $i^3 = -i$ to simplify.
<p>Expand the determinant with complex entries. Compute each 2×2 minor carefully, tracking powers of $i$. Simplify to obtain the real and imaginary parts of the result.</p>
Correct Answer: D