Question

The value of $\begin{vmatrix} x+y&y+z & z+ x \\[0.3em] x & y & z \\[0.3em] x-y & y-z & z-x \end{vmatrix}$ = is equal to :

• A. $0$
• B. $(x+y+z)^3$
• C. $2(x+y+z)^3$
• D. $2 (x+y+z)^2$

Answer 1

Answer: (A)

Let $A = \begin{bmatrix}x+y&y+z&z+x\\ x&y&z\\ x-y&y-z&z-x\end{bmatrix} $
Applying $ C_{1} \to C_{1} + C_{2} +C_{3} $
$= \begin{bmatrix}2\left(x+y+z\right)&y+z&z+x\\ x+y+z&y&z\\ 0&y-z&z-x\end{bmatrix} $
$= \left(x+y+z\right) \begin{bmatrix}2&y+z&z+x\\ 1&y&z\\ 0&y-z&z-x\end{bmatrix} $
Applying $ R_{2} \to 2R_{2} -R_{1} $
$ = \left(x+y+z\right) \begin{bmatrix}2&y+z&z+x\\ 0&y-z&z-x\\ 0&y-z&z-x\end{bmatrix}$
$= 0$ ($\because$ Two rows are identical)