Question

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

• A. $\left[\begin{array}{cc}a+ib& -c-id\\ c-id& a-ib\end{array}\right]$
• B. $\left[\begin{array}{cc}a-ib& c+id\\ -c+id& a+ib\end{array}\right]$
• C. $\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$
• D. $\left[\begin{array}{cc}a+ib& c+id\\ c-id& a-ib\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$

$A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right]$

$A^{-1}=\frac{adj\left(A\right)}{|A|}$

$|A|=(a+ib)(a-ib)-(c+id){(}^{-}c+id)$

$|A|=a^2+b^2-[-d^2-c^2]=a^2+b^2+c^2+d^2$

$|A|=1$

$M_{11}=a-ib$

$M_{12}=c-id$

$M_{21}=-c-id$

$M_{22}=a+ib$

$\Rightarrow$ cofactor matrix = $$\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$$

Adj(A) = $$\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$$