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 $\begin{vmatrix}y+z&x-z&x-y\\ y-z &z+x&y-x\\ z-y &z-x&x+y\end{vmatrix}$
Applying $C_{1} \to C_{1} + C_{2} +C_{3}$
$ = \begin{vmatrix}2x&x-z&x-y\\ 2y &z+x&y-x\\ 2z &z-x&x+y\end{vmatrix}$
Applying $ R_{1} \to R_{2}+R_{1}, R_{3} \to 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
$= 2x (4x^2) - (x - z) [4x (x + y)] + (x - y) [- 4x (x + z)] $
$= 8x^2 - (x - z) (4x^2 + 4xy) - (x - y) (4x^2 + 4xz)$
$= 8x^3 - 4x^3 - 4x^2y + 4^zx2 + 4xyz - 4x^3 - 4x^2z + 4yx^2 + 4xyz$
$= 8xyz $
Given : $ A = kxyz \ \Rightarrow \ 8xyz = kxyz \ \Rightarrow \ k = 8$