= Odd-dimensional real matrix with positive determinant has a positive eigenvalue
{title2=$A\in M_{2m+1}(\mathbb R),\ \det A>0\implies\exists\lambda>0:\det(\lambda I-A)=0$}
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> $-1$.
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