Let $n \ge 1$ and R be a unital $\mathbb{R}$-subalgebra of $M_n(\mathbb{R})$. If $A \in R \cap GL_n(\mathbb{R})$, is there any criteria to guarantee that its inverse is also in R? Since R is a vector space over the reals, this can be simplified to finding out when the adjugate of A remains in R.
I initially expected that R being closed under transposes would be sufficient, but this doesn't seem to be good enough, since the algorithm for constructing the adjugate involves taking determinants of sub-matrices of A, which might complicate things.