Question

Prove that `|[alpha,beta,gamma] ,[alpha^2,beta^2,gamma^2] , [beta+gamma, gamma+alpha, beta+alpha]|` = `(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)`

Answer 1

`L.H.S. = |[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,gamma+alpha,beta+alpha]|`
Applying `R_3->R_3+R_1`
` = |[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[alpha+beta+gamma,alpha+beta+gamma,alpha+beta+gamma]|`
` =(alpha+beta+gamma) |[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[1,1,1]|`
Applying `C_2->C_2-C_1 and C_3->C_3-C_1`
` =(alpha+beta+gamma) |[alpha,beta-alpha,gamma-alpha],[alpha^2,beta^2-alpha^2,gamma^2-alpha^2],[1,0,0]|`
` =(alpha+beta+gamma)(beta-alpha)(gamma-alpha) |[alpha,1,1],[alpha^2,(beta+alpha),(gamma+alpha)],[1,0,0]|`
` =(alpha+beta+gamma)(beta-alpha)(gamma-alpha) [gamma-alpha-beta-alpha]`
`=(alpha+beta+gamma)(beta-alpha)(gamma-alpha)(gamma-beta)`
`=(alpha+beta+gamma)(alpha-beta)(beta-gamma)(gamma-alpha) = R.H.S.`