Q. The value of $\left| \begin{matrix} a^{2} & b^{2} & c^{2} \\ (a + 1)^{2} & (b + 1)^{2} & (c + 1)^{2} \\ (a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2} \end{matrix} \right|$ is
A) $4\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
1 & 1 & 1
\end{matrix} \right|$
B) $3\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
1 & 1 & 1
\end{matrix} \right|$
C) $2\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
1 & 1 & 1
\end{matrix} \right|$
D) None of these
Explanation
(a)
$\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
(a + 1)^{2} & (b + 1)^{2} & (c + 1)^{2} \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
Applying $R_{2} \rightarrow R_{2} - R_{3}$
$= \left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
4a & 4b & 4c \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
$= 4\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
Applying $R_{3} \rightarrow R_{3} - (R_{1} - 2R_{2})$
$= 4\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
1 & 1 & 1
\end{matrix} \right|$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.