Evaluate the determinant | rr a+i b & c+i d -c+i d & a-i b | - Vidyadip

Question

Evaluate the determinant $\left|\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$

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Answer

Let $A =\left|\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$
$\Rightarrow|A|=(a+i b)(a-i b)-(c+i d)(-c+i d)$
$=(a+i b)(a-i b)+(c+i d)(c-i d)$
$=a^2-i^2 b^2+c^2-i^2 d^2$
$=a^2-(-1) b^2+c^2-(-1) d^2$
$=a^2+b^2+c^2+d^2$
Thus $, |A|=a^2+b^2+c^2+d^2$

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