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Show that |[1,alpha,alpha^2],[1,beta,bet...

Show that `|[1,alpha,alpha^2],[1,beta,beta^2],[1,gamma,gamma^2]|=(alpha-beta)(beta-gamma)(gamma-alpha)`

Text Solution

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`L.H.S. = |[1,alpha,alpha^2],[1,beta,beta^2],[1,gamma,gamma^2]|`
Applying `R_1->R_1-R_3` and `R_2->R_2-R_3`
`= |[0,alpha-gamma,alpha^2-gamma^2],[0,beta-gamma,beta^2-gamma^2],[1,gamma,gamma^2]|`
`=(alpha-gamma)(beta-gamma)|[0,1,alpha+gamma],[0,1,beta+gamma],[1,gamma,gamma^2]|`
`=(alpha-gamma)(beta-gamma)[beta+gamma - alpha-gamma]`
`= (alpha-gamma)(beta-gamma)(beta-alpha)`
`=(-1)(-1)(alpha-beta)(beta-gamma)(gamma-alpha)`
`=(alpha-beta)(beta-gamma)(gamma-alpha) = R.H.S.`
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