R1 R1 - R2 (i.e. Replacing 1st row by subtraction of 1st and 2nd row)

R2 R2 - R3 (i.e. Replacing 2nd row by subtraction of 2nd and 3rd row)



Since we know a2 - b2 = (a + b)(a - b)


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



R1 R1 - R2 (i.e. Replacing 1st row by subtraction of 1st and 2nd row)



Expanding the determinant along 1st column



LHS = (a - b)(b - c)(0 - (a - c))


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



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