Explanation

(c)
Given, [λ($\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}})\ \lambda^{2}\overrightarrow{\mathbf{b}}\ \ \ \lambda\overrightarrow{\mathbf{c}}\rbrack = \lbrack\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{b}}\rbrack$
$\Longrightarrow \ \left| \begin{matrix}
\lambda\left( a_{1} + b_{1} \right) & \lambda\left( a_{2} + b_{2} \right) & \lambda\left( a_{3} + b_{3} \right) \\
\lambda^{2}b_{1} & \lambda^{2}b_{2} & \lambda^{2}b_{3} \\
\lambda c_{1} & \lambda c_{2} & \lambda c_{3}
\end{matrix} \right| = \left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} + c_{1} & b_{2} + c_{2} & b_{3} + c_{3} \\
b_{1} & b_{2} & b_{3}
\end{matrix} \right|$
$\Longrightarrow \lambda^{4}\left| \begin{matrix}
a_{1} + b_{1} & a_{2} + b_{2} & a_{3} + b_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right| = \left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} + c_{1} & b_{2} + c_{2} & b_{3} + c_{3} \\
b_{1} & b_{2} & b_{3}
\end{matrix} \right|$
[applying $R_{1} \rightarrow R_{1} - R_{2}$ in LHS and $R_{2} \rightarrow R_{2} - R_{3}$ in RHS]
$\Longrightarrow \lambda^{4}\left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right| = - \left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right|$
$\Longrightarrow \ \lambda^{4} = - 1$
Hence, no real value of λ exists.