Applying the row operations \(R_3 \to R_3 - R_2\) and then \(R_2 \to R_2 - R_1\) reduces the rows to \((1/a, bc, a^3)\), \((1/b, ca, b^3)\) and \((1/c, ab, c^3)\). Scaling these rows by \(a, b, c\) respectively makes the second column exactly \(abc\) times the first column, so the two columns become proportional. Hence the determinant equals 0.
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