Let \(M = \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{bmatrix}\) where \(a, b\) and \(c\) are real numbers such that \(a + b + c = 0\) and \(abc \neq 0\). If \(\det M = 0\), then…

The maximum is $5$. The determinant equals $(b-a)(c-a)(c-b)$, so at least two variables coincide. If $a=b$, then $c=-2a$ and the ratio is $\frac12$, and if $a=c$ or $b=c$, the remaining variable is $-2c$ and the ratio is $5$. The value $4$ comes from dropping the smaller squared term in the latter cases.