Explanation

(c)
We have,
$A^{2} = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} = \begin{bmatrix}
a^{2} + bc & ab + db \\
ac + cd & bc + d^{2}
\end{bmatrix}$
As A satisfies $x^{2} + k = 0,$ therefore
$A^{2} + kI = O$
$\Rightarrow \begin{bmatrix}
a^{2} + b + k & (a + d)b \\
(a + d)c & bc + d^{2} + k
\end{bmatrix} = \begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}$
$\Rightarrow a^{2} + bc + k = 0,\ bc + d^{2} + k = 0$
and $(a + d)b = (a + d)c = 0$
As $bc \neq 0,\ b \neq 0,\ c \neq 0,$ so
$a + d = 0$
$\Rightarrow a = - d$
Also,
$k = - \left( a^{2} + bc \right)$
$= - \left( d^{2} + bc \right)$
$= - \left( ( - ad) + bc \right)$
$= |A|$