Complexity of class finding selection of entries in matrix
Let $U$ be the set of rows in the given matrix and $V$ the set of columns. For each $x$ at row $i$ and column $j$, add an edge between row $i$ and column $j$. You will have an (unweighted) bipartite graph $G=(U,V, E)$. The problem is to find a perfect matching, which is basically the same as find the maximum matching of $G$.
There are various algorithms that compute the maximum cardinality matching. For example, The Hopcroft–Karp algorithm runs in $O(\max(|E|\sqrt {|V|}, |V|^2)$.
So, we can say that the complexity class of the problem in the question is in $O(n^{2.5})$ since $|E|\le n^2$ and $|V|= 2n$.