Show that the matrix `A = [[1 , a,alpha , aalpha],[1, b, beta, b beta ],[1 ,c,gamma ,cgamma ]]` is of renk 3
provided no two of a, b, c are equal and no two of `alpha ,beta,gamma `
are equal.

Explanation

We have , `A= [[1,a, alpha , aalpha],[1, b ,beta,b beta ],[1 ,c, gamma,cgamma]]`
Applying `R_(2) rarr R_(2) - R_(1) and R_(3) rarr R_(3) -R_(1),` we get
`A= [[1,a, alpha , aalpha],[0, b-a ,beta-alpha,b beta-aalpha ],[0,c-a, gamma-alpha,cgamma-aalpha]]`
Applying `C_(2) rarr C_(2) - aC_(1), C_(3) rarr C_(3)- alphaC_(1) and C_(4) rarr C_(4) - a alpha C_(1), ` we get
`A= [[1,0, 0 , 0],[0, b-a ,beta-alpha,b beta-aalpha ],[0,c-a, gamma-alpha,cgamma-aalpha]]`
Applying ` C_(4) rarr C_(4) - alpha C_(2) - bC_(3)` we get
`A= [[1,0, 0 , 0],[0, b-a ,beta-alpha,0 ],[0,c-a, gamma-alpha,(c-b)(gamma-alpha)]]`
For `p(A)= 3`
`c- a ne 0 , gamma - alpha ne 0, c-bne 0, b-a ne0 , beta - alpha ne 0 `
i.e.,`ane b, bnec, cnea and alpha ne beta, beta ne gamma , gamma ne alpha `