$A \in M_2(\mathbb{R})$ positive stable if and only if $\textrm{trace}(A) > 0$ and $\det(A) > 0$?

Question

Why is a real-valued matrix say $A \in M_2(\mathbb{R})$ "positive stable", if and only if $\textrm{trace}(A) > 0$ and $\det(A) > 0$?

Thank you so much

Answer 1

If both eigenvalues of $A$ are real, the trace and the determinant are both positive if and only if both eigenvalues are positive.

If the eigenvalues of $A$ are non-real, they must form a conjugate pair. and $\det(A)$ is automatically positive. Since the trace of $A$ is the double of the real parts of either eigenvalue, the conclusion follows.