Question

The number of matrices $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{ (where }a,b,c,d\in R\text{ ) }$ such that $A^{-1}=-A$ is :

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (D)

As $A^{-1}$ exists, $\det (A)=ad-bc\ne 0$.
Also, $\det (-A)=\det \left[\begin{array}{cc}-a& -b\\ -c& -d\end{array}\right]=ad-bc$
$=\det (A)$
since $\det \left(A^{-1}\right)=\frac{1}{\det (A)},A^{-1}=-A$ implies
$\frac{1}{ad-bc}=ad-bc$
$\Rightarrow (ad-bc{)}^2=1\Rightarrow ad-bc=\pm 1$.
If $ad-bc=1$, then $A^{-1}=-A$
gives
$A^{-1}=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]=\left[\begin{array}{cc}-a& -b\\ -c& -d\end{array}\right]$
$\Rightarrow a+d=0$
$\therefore a(-a)-bc=1$
$\Rightarrow bc=-\left(1+a^2\right)\Rightarrow bc\ne 0\text{ and }$
$b=-\frac{1+a^2}{c}$
$\therefore A=\left[\begin{array}{cc}a& -\frac{\left(1+a^2\right)}{c}\\ c& -a\end{array}\right]\text{, where }c\ne 0\text{. }$
Thus, there are infinite number of such matrices.