Question

Evaluate the following determinants,
$\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-id\end{array}\right|$

• A. ${\left(a+b\right)}^2$
• B. ${\left(a+b+c+d\right)}^2$
• C. $\left(a^2+b^2-c^2-d^2\right)$
• D. $\left(a^2+b^2+c^2+d^2\right)$

Answer 1

Answer: (D)

We have, $\left|\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right|$
$=\left(a+ib\right)\left(a-ib\right)-\left(c+id\right)\left(-c+id\right)$
$=\left(a^2+b^2\right)-\left(-c^2-d^2\right)=a^2+b^2+c^2+d^2$