Q. The value of $\left| \begin{matrix} x + y & y + z & z + x \\ x & y & z \\ x - y & y - z & z - x \end{matrix}\ \right|\ is\ equal\ to$
A) $2(x + y + z)^{2}$
B) $2(x + y + z)^{3}$
C) $(x + y + z)^{3}$
D) 0
Explanation
(d)
Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3},$ we get
$\left| \begin{matrix}
2(x + y + z) & y + z & z + x \\
x + y + z & y & z \\
0 & y - z & z - x
\end{matrix} \right|$
$= (x + y + z)\left| \begin{matrix}
2 & y - z & z + x \\
1 & y & z \\
0 & y - z & z - x
\end{matrix} \right|$
Applying $R_{2} \rightarrow 2R_{2} - R_{1}$
$= (x + y + z)\left| \begin{matrix}
2 & y + z & z + x \\
0 & y - z & z - x \\
0 & y - z & z - x
\end{matrix} \right|$
$= 0\ \ \ \lbrack\therefore two\ rows\ are\ identical\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.