Question

Write the value of  \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]

Answer 1

\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\] 
\[ = a^2 - iab + iab - i^2 b^2 - ( - c^2 - icd + icd + i^2 d^2 )\] 
\[ = a^2 - i^2 b^2 + c^2 - i^2 d^2 \] 
Here,
\[ i^2 = - 1\] 
\[ \Rightarrow \begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} = a^2 + b^2 + c^2 + d^2 \]