Q74 / 180 · COMEDK 2016 · Mathematics

The value of $\left|\begin{array}{ccc}x+y & y+z & z+x \\ x & y & z \\ x-y & y-z & z-x\end{array}\right|=$

  • Correct answer
    A
    0
  • B
    $(x+y+z)^{3}$
  • C
    $2(x+y+z)^{3}$
  • D
    $2(x+y+z)^{2}$

Solution

$\left|\begin{array}{ccc} x+y & y+z & z+x \\ x & y & z \\ x-y & y-z & z-x \end{array}\right|$ $\begin{aligned}=&\left|\begin{array}{ccc}2(x+y+z) & y+z & z+x \\ (x+y+z) & y & z \\ 0 & y-z & z-x\end{array}\right| \\ & \quad\left[\because C_{1} \rightarrow C_{1}+C_{2}+C_{3}\right] \end{aligned}$ $=(x+y+z)\left|\begin{array}{lcc}2 & y+z & z+x \\ 1 & y & z \\ 0 & y-z & z-x\end{array}\right|$ $=(x+y+z)\left|\begin{array}{lll}2 & y+z & z+x \\ 0 & y-z & z-x \\ 0 & y-z & z-x\end{array}\right|$ $\left[\because R_{i_{2}} \rightarrow 2 R_{2}-R_{1}\right]$ $=(x+y+z) \times 0=0$ $\left[\because R_{2}\right.$ and $R_{3}$ are identical $]$

Hence, the correct answer is option: 'A'.

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