Let $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the real roots of the equation ${\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0 (\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R}\text{ and }\mathrm{a}\ne 0)$. If the system of equations (in u, v and w) given by

$\begin{array}{l}\mathrm{\alpha u}+\mathrm{\beta v}+\mathrm{\gamma w}=0\\ \mathrm{\beta u}+\mathrm{\gamma v}+\mathrm{\alpha w}=0\\ \mathrm{\gamma u}+\mathrm{\alpha v}+\mathrm{\beta w}=0\end{array}$

has non-trivial solutions, then the value of a²/b is _______.

Equation ${\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ has roots $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$. Therefore,

$\begin{array}{l}\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=-\mathrm{a}\\ \mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha }=\mathrm{b}\end{array}$

Since the given system of equations has non-trivial solutions, we have

$\left|\begin{array}{ccc}\mathrm{\alpha }& \mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\beta }& \mathrm{\gamma }& \mathrm{\alpha }\\ \mathrm{\gamma }& \mathrm{\alpha }& \mathrm{\beta }\end{array}\right|=0$

$\begin{array}{l}\mathrm{or} {\mathrm{\alpha }}^3+{\mathrm{\beta }}^3+{\mathrm{\gamma }}^3-3\mathrm{\alpha \beta \gamma }=0\\ \mathrm{or} (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\left[{\mathrm{\alpha }}^2+{\mathrm{\beta }}^2+{\mathrm{\gamma }}^2-\mathrm{\alpha \beta }-\mathrm{\beta \gamma }-\mathrm{\gamma \alpha }\right]=0\\ \mathrm{or} (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\left[(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }{)}^2-3(\mathrm{\alpha \beta }+\mathrm{\beta \gamma }+\mathrm{\gamma \alpha })\right]=0\\ \Rightarrow -\mathrm{a}\left[{\mathrm{a}}^2-3\mathrm{b}\right]=0\\ \mathrm{or} {\mathrm{a}}^2/\mathrm{b}=3\end{array}$