Question

If $\left|\begin{array}{ccc}y+z& x-z& x-y\\ y-z& z+x& y-x\\ z-y& z-x& x+y\end{array}\right|=kxyz$,
then the value of $k$ is :

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (D)

Let $A=\left|\begin{array}{ccc}y+z& x-z& x-y\\ y-z& z+x& y-x\\ z-y& z-x& x+y\end{array}\right|$
Applying $C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}2x& x-z& x-y\\ 2y& z+x& y-x\\ 2z& z-x& x+y\end{array}\right|$
Applying $R_1\to R_2+R_1,R_3\to R_3+R_1$
$=\left|\begin{array}{ccc}2x& x-z& x-y\\ 2\left(x+y\right)& 2x& 0\\ 2\left(z+z\right)& 0& 2x\end{array}\right|$
On expanding, we get
$=2x\left(4x^2\right)-(x-z)[4x(x+y)]+(x-y)(-4x(x+z)]$
$=8x^2-(x-z)\left(4x^2+4xy\right)-(x-y)\left(4x^2+4xz\right)$
$=8x^3-4x^3-4x^{2}y+4z{x}^2+4xyz-4x^3-4x^{2}z+4y{x}^2+4xyz=8xyz$
Given: $A=kxyz$
$\Rightarrow 8xyz=kxyz$
$\Rightarrow k=8$