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|(x,x^2,y2),(y,y^2,2x),(z,z^2,xy)| = (x-...

`|(x,x^2,y2),(y,y^2,2x),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+2x)`

Text Solution

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`L.H.S. = |[x,x^2 ,yz] ,[y , y^2, zx],[z,z^2,xy]|`
Operating `R_1->R_1- R_2 and R_2->R_2-R_3`
`=|[x-y,x^2-y^2,yz-zx],[y-z,y^2-z^2,zx-xy],[z,z^2,xy]|`
`=|[x-y,(x-y)(x+y),yz-zx],[y-z,(y-z)(y+z),zx-xy],[z,z^2,xy]|`
`=(x-y)(y-z)|[1,x+y,-z],[1,y+z,-x],[z ,z^2,xy]|`
Now, operating `R_1->R_1-R_2`
`=(x-y)(y-z)|[0,x-z,x-z],[1,y+z,-x],[z ,z^2,xy]|`
`=(x-y)(y-z)(z-x)|[0,-1,-1],[1,y+z,-x],[z ,z^2,xy]|`
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