Using properties of determinants, prove the following: \begin{vmatrix} \alpha & \beta & \gamma \\ \alpha ^{2}& \beta ^{2} &\gamma ^{2} \\ \beta +\gamma & \gamma +\alpha & \beta +\alpha \end{vmatrix}`=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)`

Explanation

`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]`

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