Question

Write the value of $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|.$

Answer 1

$$\begin{gathered} \left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right| \\ =a^2-\cancel{iab}+\cancel{iab}-i^2{b}^2-(-c^2-\cancel{icd}+\cancel{icd}+i^2{d}^2) \\ =a^2-i^2{b}^2+c^2-i^2{d}^2 \\ \mathrm{Here}, \\ {i}^2=-1 \\ \Rightarrow \left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|=a^2+b^2+c^2+d^2 \end{gathered}$$