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Prove that : Det[[x,x^2,x^3],[y,y^2,y^3]...

Prove that : `Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-x)`

Text Solution

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`L.H.S. = |[x,x^2 ,x^3] ,[y , y^2, y^3],[z,z^2,z^3]|`
`= xyz|[1,x ,x^2] ,[1 , y, y^2],[1,z,z^2]|`
Operating `R_2->R_1- R_2 and R_3->R_1-R_3`
`= xyz|[1,x ,x^2] ,[0 ,x- y, x^2-y^2],[0,x-z,x^2-z^2]|`
`= xyz|[1,x ,x^2] ,[0 ,x- y, (x-y)(x+y)],[0,x-z,(x-z)(x+z)]|`
`= xyz(x-y)(x-z)|[1,x ,x^2] ,[0 ,1, x+y],[0,1,x+z]|`
`= xyz(x-y)(x-z) [x+z-x-y]`
`= xyz(x-y)(x-z)(z-y)`
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