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 $\left|\begin{array}{ccc}y+z& x-z& x-y\\ y-z& z+x& y-x\\ z-y& z-x& x+y\end{array}\right|$
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
$=2x(4x^2)-(x-z)[4x(x+y)]+(x-y)[-4x(x+z)]$
$=8x^2-(x-z)(4x^2+4xy)-(x-y)(4x^2+4xz)$
$=8x^3-4x^3-4x^{2}y+4^{z}x2+4xyz-4x^3-4x^{2}z+4y{x}^2+4xyz$
$=8xyz$
Given : $A=kxyz\Rightarrow 8xyz=kxyz\Rightarrow k=8$