Question

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

• A. $a+d=0$
• B. $k=-|A|$
• C. $k=a^2+b^2+c^2+d^2$
• D. $k=|A|$

Answer 1

Answer: (A), (D)

The correct options are
A $a+d=0$
D $k=|A|$

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ satisfies the equation $x^2+k=0$
Characteristic eqaution
$AX=\lambda X$
$\Rightarrow |A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}a-\lambda & b\\ c& d-\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^2-(a+d)\lambda +ad-bc=0$
comparing the above equation with $x^2+k=0$ gives
$a+d=0$ and $k=ad-bc=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=|A|$
Hence, option A and D.