Question

If $A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ 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)

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