Question

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

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

Answer 1

Answer: (C)

$k=ad-bc$

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

Consider $|A-\lambda I|=0$

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

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

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

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

Compare it with given equation

$k=ad-bc$