Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ such that $A$ satisfies the relation $A^2-(a+d)A=0$, then inverse of $A$ is

• A. $I$
• B. $A$
• C. $(a+d)A$
• D. none of these

Answer 1

The correct option is D none of these

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ such that $A$ satisfies the relation $A^2-(a+d)A=0$
$\Rightarrow A^2-(a+d)A=0$
$\Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]-(a+d)\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]-(a+d)\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow a^2+bc-a^2-ad=0$
$\Rightarrow ad-bc=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=0$
$\therefore$ Inverse of A doesnot exist.
Hence, option D.