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

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

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

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

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

Expanding the determinant along 1^st column

∴

∴LHS = (a - b)(b - c)(0 - (a - c))

∴LHS = (a - b)(b - c)(c - a) = RHS

∴