 Odd-dimensional real matrix with positive determinant has a positive eigenvalue

For a real square matrix $A$ of odd size, if $\det A>0$, then $A$ has a positive real eigenvalue. Its characteristic polynomial $p(t)=\det (tI-A)$ is monic of odd degree, so $p(0)=-\det A<0$ and $p(t)\to +\infty$ as $t\to +\infty$. The intermediate value theorem supplies a positive root. This proves the claim without assuming that all eigenvalues are real. The odd-size hypothesis is essential: $-I$ in even dimension has positive determinant and only the negative eigenvalue $-1$.