Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is such that $|A|=0$ and $A^2-(a+d)A+kI=0$, then $k$ is equal to

• A. $b+c$
• B. $a+d$
• C. $ab+cd$
• D. none of these.

Answer 1

The correct option is D none of these.

As $|A|=0$, we get $ad-bc=0$
Also, $A^2-(a+d)A=A[A-(a+d\left)I\right]$
$=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}-d& b\\ c& -a\end{array}\right]=\left[\begin{array}{cc}-ad+bc& ab-ba\\ -cd+cd& bc-ad\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Thus, $kI=0\Rightarrow k=0$