सिद्ध कीजिए - `|(3k,-x+y,-x+z),(x-y,3y,z-y),(x-z,y-z,3z)|=3(x+y+z)(xy+yz+zx)`.

Explanation

L.H.S.`=|(3k,-x+y,-x+z),(x-y,3y,z-y),(x-z,y-z,3z)|`
`=|(x+y+z,-x+y,-x+z),(x+y+z,3y,z-y),(x+y+z,y-z,3z)|`
संक्रिया `C_1toC_1+C_2+C_3` से)
`=(x+y+z)=|(1,-x+y,-x+z),(1,3y,z-y),(1,y-z,3z)|`
[`C_1` से `(x+y+z)` उभयनिष्ट लेने पर]
`=(x+y+z)|(1,-x+y,-x+z),(0,2x+x,x-y),(0,x-z,2z+x)|`
(संक्रियाओं `R_2toC_2-R_2` और `R_3toR_3-R_1` से)
`=(x+y+z)xx1xx|(2y+x,x-y),(x-z,2z+x)|`
`=(x+y+z)[(2y+x)(2z+x)-(x-y)(x-z)]`
`=(x+y+z)[(4yz+2xy+2xy+2zx+x^2-x^2+xz+xy-yz]`
`=(x+y+z)[(3xy+3yz+3zx]`
`=3(x+y+z)(xy+yz+zx)`
R.H.S.