Answer :

C1 C1 - C2 (i.e. Replacing 1st column by subtraction of 1st and 2nd column)

C2 C2 - C3 (i.e. Replacing 2nd column by subtraction of 2nd and 3rd column)



We have a3 - b3 = (a - b)(a2 + ab + b2) and b3 - c3 = (b - c)(b2 + bc + c2)


Therefore taking (a - b) and (b - c) outside the determinant from 1st and 2nd column respectively



C1 C1 - C2 (i.e. Replacing 1st column by subtraction of 1st and 2nd column)



As a2 - c2) = (a + c)(a - c) therefore taking (a - c) outside the determinant from 1st column we get


.


Expanding the determinant along 1st row


LHS = (a - b)(b - c)(a - c)( - (a + b + c))


Adjusting the minus sign with (a - c)


LHS = (a - b)(b - c)(c - a)(a + b + c) = RHS



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