Question

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

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

Answer 1

Answer: (D)

Suppose $A^{-1}$ exists, then
$A^{-1}{A}^2-(a+d)A^{-1}A=O$
$\Rightarrow A-(a+d)I=O$
$\Rightarrow \left[\begin{array}{cc}-d& b\\ c& -a\end{array}\right]=O$
$\Rightarrow a=b=c=d=0\Rightarrow |A|=0$
$A$ contradiction. Thus, $A^{-1}$ does not exist.