आव्यूह `|{:(x,y,y+z),(z,x,z+x),(y,x,x+y):}|` का सारणिक कौन-सा है ?

Options

A. `(x-y)(y-z)(z-x)`
B. `(x-y)(y-z)`
C. `(y-z)(z-x)`
D. `(z-x)^(2)(x+y+z)`

Explanation

दिया है
`|{:(x,y,y+z),(z,x,z+x),(y,z,x+y):}|`
`[R_(1) rarrR_(1) + R_(2) + R_(3) "से"]`
`|{:(x+y+z,x+y+z,2(x+y+z)),(z,x,z+x),(y,z,x+y):}|`
`= (x+y+z) |{:(1,1,2),(z,x,z+x),(y,z,x+y):}|`
`[C_(2) rarr C_(2) - C_(1) " तथा" C_(3) rarr C_(3) - 2C_(1) "से"]`
`= (x+y+z) |{:(1,0,0),(z,x-z,x-z),(y,z-y,x-y):}|`
`= (x+y+z) [1(x-z)(x-y) - (x-z) (z-y)]`
`= (x+y+z) [(x-z)(x-y-z+y]`
`= (x+y+z) (x-z) (x-z)`
`= (x+y+z) (z-x)^(2)`