`" if " Delta = |{:(abc,,b^(2)c,,c^(2)b),(abc ,,c^(2)a,,ca^(2)),( abc,,a^(2)b,,b^(2)a):}| =0 , (a, b, c in R " and are all "`
different and non- zero ) the prove that `a+b+c=0`

Explanation

`Delta =|{:(abc,,b^(2)c,,c^(2)b),(abc,,c^(2)a,,ca^(2)),(abc,,a^(2)b,,b^(2)a):}|=0`
Taking bc , ca and ab common from `R_(1),R_(2)" and " R_(3)` respectively we get
`:. (bc.ca.ab) |{:(a,,bc,,c),(b,,c,,a),(c,,a,,b):}|=0`
`rArr |{:(a,,bc,,c),(b,,c,,a),(c,,a,,b):}|=0`
`rArr -(a^(3)+b^(3)+c^(3)-3abc)=0`
`rArr (1)/(2) (a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]=0`
`rArr a+b+c=0`