Suppose \(\left|\begin{array}{lll}a & a^{2} & a^{3}-1 \\ b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right|=0\), where \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) are distinct real…

The product $abc$ is $1$. Column dependence gives a nonzero polynomial $u x+v x^2+w(x^3-1)$ vanishing at $a,b,c$, and three distinct roots force $w\ne0$. Dividing by $w$ gives a monic cubic with constant term $-1$, which equals $-abc$, while dropping the minus sign on the product gives $-1$.