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