Evaluate: a + ib & c + id -c + id & a - ib - Vidyadip

Question

$\text{Evaluate:} \begin{vmatrix} a + ib & c + id \\ -c + id & a - ib \\ \end{vmatrix} $

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Answer

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

$= (a + ib) (a - ib) - (c + id) (- c + id)$

$= [a^{2} - i^{2}b^{2} ] - [i^{2} b^{2} - c ^{2} ]$

$= (a^{2} + b ^{2} ) - (- d^{2} - c ^{2} )$

$= a^{2} + b^{2} + c^{2}+ d^{2}$

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