Question

The number of $2\times 2$ matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ for which ${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}$ $=\left[\begin{array}{cc}\frac{1}{a}& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{d}\end{array}\right]$, $(a,b,c,dϵR)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $Infinite$

Answer 1

The correct option is A $0$

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]{\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\left[\begin{array}{cc}1/a& 1/b\\ 1/c& 1/d\end{array}\right]$ Not possible for any values of $a,b,c$

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=Adj\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]\Rightarrow A^{-1}=\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right]$