prove that `[ [a-b-c , 2a , 2a ] , [2b , b-c-a , 2b ] ,[2c ,2c,c-a-b]]`= `(a+b+c)^3`

Explanation

Applying `C_(1) to C_(1) -C_(2) " and "C_(2) to C_(2)-C_(3)` and taking (a+b+c) common from each of `C_(1) " and "C_(2)` we get
`D= (a+b+c)^(2) xx |{:(-1 ,,0,,2a),(1,,-1,,2a),(0,,1,,c-a-b):}|`
`" Now " R_(3) to R_(3) + R_(2)` gives
`D= (a+b+c)^(2) xx |{:(-1 ,,0,,2a),(1,,-1,,2a),(0,,0,,(c+a+b)):}|`
Expanding along `R_(3)` we get
`D=(a+b+c)^(2) (a+b+c) =(a+b+c)^(3)`