Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ satisfies the equation $x^2-(a+d)x+k=0$ then

• A. $k=bc$
• B. $k=ad$
• C. $k=a^2+b^2+c^2+d^2$
• D. $k=ad-bc$

Answer 1

Answer: (D)

$k=ad-bc$

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$⟹\left|\begin{array}{cc}\lambda -a& b\\ c& \lambda -d\end{array}\right|=0$

$(\lambda -a)(\lambda -d)-bc=0$

$⟹{\lambda }^2-(a+d)\lambda +(ad-bc)=0$

Comparing with $x^2-(a+d)x+k=0$

$⟹k=ad-bc$