Question

If $\begin{vmatrix}y+z&x-z&x-y\\ y-z&z+x&y-x\\ z-y&z-x&x+y\end{vmatrix} =kxyz$,
then the value of $k$ is :

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (D)

Let $A=\begin{vmatrix}y+z&x-z&x-y\\ y-z&z+x&y-x\\ z-y&z-x&x+y\end{vmatrix} $
Applying $C_{1}\rightarrow C_{1}+C_{2}+C_{3}$
$=\begin{vmatrix}2x&x-z&x-y\\ 2y&z+x&y-x\\ 2 z&z-x&x+y\end{vmatrix}$
Applying $R_{1}\rightarrow R_{2}+R_{1}, R_{3}\rightarrow R_{3}+R_{1} $
$=\begin{vmatrix}2x&x-z&x-y\\ 2\left(x+y\right)&2x&0\\ 2 \left(z+z\right)&0&2x\end{vmatrix}$
On expanding, we get
$=2 x\left(4 x^{2}\right)-(x-z)[4 x(x+y)]+(x-y)(-4 x(x+z)] $
$=8 x^{2}-(x-z)\left(4 x^{2}+4 x y\right)-(x-y)\left(4 x^{2}+4 x z\right)$
$= 8 x^{3}-4 x^{3}-4 x^{2} y+4 z x^{2}+4 x y z-4 x^{3}-4 x^{2} z+4 y x^{2}+4 x y z =8 x y z$
Given: $A=kxyz$
$ \Rightarrow 8 xyz=k x y z$
$ \Rightarrow k=8$