  For a field $R$ of characteristic $p\ge 0$ and a matrix $c$ in the full
$n\times n$ matrix algebra $M_n(R)$ over $R$, let $S_n(c,R)$ be the centralizer
algebra of $c$ in $M_n(R)$. We show that $S_n(c,R)$ is a Frobenius-finite,
$1$-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension
$S_n(c,R)\subseteq M_n(R)$ is separable and Frobenius. Further, we study the
isomorphism problem of invariant matrix algebras. Let $\sigma$ be a permutation
in the symmetric group $\Sigma_n$ and $c_{\sigma}$ the corresponding
permutation matrix in $M_n(R)$. We give sufficient and necessary conditions for
the invariant algebra $S_n(c_{\sigma},R)$ to be semisimple. If $R$ is an
algebraically closed field, we establish a combinatoric characterization of
when two semisimple invariant $R$-algebras are isomorphic in terms of the cycle
types of permutations.
