सारणिकों के गुणधर्मों का प्रयोग करके निम्नलिखित प्रश्न को सिद्ध कीजिए।
`|(x,x^(2),1+px^(3)),(y,y^(2),1+py^(3)),(z,z^(2),1+pz^(3))|`
`=(1+pxyz)(x-y)(y-z)(z-x)`

Explanation

L.H.S `=|(x,x^(2),1+px^(3)),(y,y^(2),1+py^(3)),(z,z^(2),1+pz^(3))|`
`=|(x,x^(2),1),(y,y^(2),1),(z,z^(2),1)|+|(x,x^(2),px^(3)),(y,y^(2),py^(3)),(z,z^(2),pz^(3))|`
`=-|(x^(2),x,1),(y^(2),y,1),(z^(2),z,1)|+pxyz|(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(2))|`
(`C_(1)hArrC_(2)` पहले सार‌ण‌िक में)
`=|(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(2))|+pxyz|(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(3))|`
(`C_(1)hArrC_(3)` पहले सार‌‌ण‌िक में
`=(1+pxyz)|(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(2))|`
`=(1+pxyz)|(1,x,x^(2)),(0,y-x,y^(2)-x^(2)),(0,z-x,z^(2)-x^(2))|`
`(R_(2)toR_(2)-R_(1),R_(3)toR_(3)-R_(1))`
`=(1+pxyz)(y-x)(z+x)|(1,x,x^(2)),(0,1,y+x),(0,1,z+x)|`
`=-(1+pxyz)(x-y)(z-x)1.|(1,y+x),(1,z+x)|`
`=-(1+pxyz)(x-y)(z-x)(z+x-y-x)`
`=(1+pxyz)(x-y)(y-z)(z-x)=R.H.S`