Show that `[[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]=0` where `alpha,beta,gamma` are in AP

Explanation

We have given : `[[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]=0`

And `alpha, beta, gamma in A.P.`

Let`Delta=[[x-3,x-4,x-alpha],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]`

Now apply `R_1->(R_1+R_2+R_3)`

`Delta=[[3x-6,3x-9,3x-(alpha+beta+gamma)],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]`

As we have given that,`alpha, beta, gamma in A.P.`

Then this `implies alpha+gamma=2beta`

`Delta=[[3(x-2),3(x-3),3(x-beta)],[x-2,x-3,x-beta],[x-1,x-2,x-gamma]]`

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