Question

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

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (A)

If $ad-bc\ne 0$, then
$A^{-1}=\frac{1}{ad-bc}\left|\begin{array}{cc}d& -b\\ -c& a\end{array}\right|$
Thus, $\frac{d}{ad-bc}=\frac{1}{a}\iff ad=ad-bc$
$\iff bc=0\iff b=0\text{ or }c=0$
Therefore,
${\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}^{-1}=\left|\begin{array}{cc}1/a& 1/b\\ 1/c& 1/d\end{array}\right|$
is never possible.