Question

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

$|A-\lambda I|=0\Rightarrow \left|\begin{array}{cc}a-\lambda & b\\ c& d-\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^2-\lambda (a+d)+ad-bc=0$
This is characteristic equation. Comparing with given equation we get
$k=ad-bc=|A|,a+d=0$