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I have a closed loop transfer function $$G_{\rm cl}(s) = \frac{k}{s^{2} + s + k}$$ I am trying to find the critical gain of the system. From using the Routh Hurwitz criterion I get a $k = 0$.

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Second order Routh Hurwitz:

$$a_0s^2+a_1s+a_2 = 0$$

$$\begin{cases} s^2 & a_0 & a_2\\ s^1 & a_1 & 0\\ s^0 & a_2\\ \end{cases}$$

In your example

$$\begin{cases} s^2 & 1 & k\\ s^1 & 1 & 0\\ s^0 & k\\ \end{cases}$$

Hence, $k$ should be positive.

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