Mathematics - Determinant Question with Solution | TestHub

Let the given determinant be $\Delta$. Apply row operations $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$:$$\Delta =\left|\begin{array}{ccc}1& a& a^3\\ 0& b-a& b^3-a^3\\ 0& c-a& c^3-a^3\end{array}\right|$$ Expand along the first column:$$\Delta =(b-a)(c^3-a^3)-(c-a)(b^3-a^3)$$ Using the identity $x^3-y^3=(x-y)(x^2+xy+y^2)$:$$\Delta =(b-a)(c-a)(c^2+ca+a^2)-(c-a)(b-a)(b^2+ba+a^2)$$ Factor out $(b-a)(c-a)$:$$\Delta =(b-a)(c-a)[(c^2+ca+a^2)-(b^2+ba+a^2)]$$ $$\Delta =(b-a)(c-a)[c^2-b^2+ca-ba]$$ $$\Delta =(b-a)(c-a)[(c-b)(c+b)+a(c-b)]$$ Factor out $(c-b)$: $$\Delta =(b-a)(c-a)(c-b)(a+b+c)$$ Rearranging factors to the standard form $(a-b)(b-c)(c-a)$:$$\Delta =(a-b)(b-c)(c-a)(a+b+c)$$