I know that $\mid A\mid$ cannot be zero.
If $A$ is non-negative, then $x^{T}Ax\geq 0$. $A$ being positive is equivalent to having all its corners as positive. The largest corner $A$ can have is $A$ itself, so therefore $\mid A\mid$ must be positive too. Does it then follow that it a non negative matrix cannot have an inverse $A^{-1}$, because it would break multiplying determinants?
I am pretty lost here, help needed if possible.