= Solution
Let $p(t)=\det(tI-A_\alpha)$ be the real <characteristic polynomial>. It is a monic cubic, so $p(t)\to+\infty$ as $t\to+\infty$. At zero,
$$
p(0)=\det(-A_\alpha)=-\det A_\alpha<0.
$$
The <intermediate value theorem> therefore gives some $t_0>0$ with $p(t_0)=0$. A root of the <characteristic polynomial> is an <eigenvalue>, hence \b[$A_\alpha$ has a positive real <eigenvalue> for every real $\alpha$]. This uses the general fact that an <odd-dimensional real matrix with positive determinant has a positive eigenvalue>.
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