Question

If $a,b,c$ and $d$ are real numbers such that $a^2+b^2+c^2+d^2=1$ and if $A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right],$where $i^2=-1,$ then $A^{-1}=$

Answer 1

$A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right]$
$A^{-1}=\frac{\text{Adj}A}{|A|}$
$\text{Adj}A=\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$
$|A|=a^2+b^2+c^2+d^2=1$
$\Rightarrow A^{-1}=\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$