Question

Evaluate the following determinant:

\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]

Answer 1

\[∆ = a^2 - i^2 b^2 - \left( i^2 d^2 - c^2 \right)\]
\[ = a^2 - i^2 b^2 - i^2 d^2 + c^2 \]
\[ = a^2 + c^2 - i^2 \left( b^2 + d^2 \right) \left[ \because i^2 = - 1 \right]\]
\[ = a^2 + c^2 + b^2 + d^2\]