Selecting k rows and k columns from a matrix to maximize the sum of the k^2 elements
There is an easy reduction from the well-known NP-complete problem MAX-CLIQUE to your problem. Recall that in MAX-CLIQUE we are given a graph $G$ and an integer $k$, and have to decide whether $G$ contains a $k$-clique. We can recast this as an instance of your problem as follows: the matrix $A$ is the adjacency matrix of your graph, and the goal is to determine whether there is a choice of indices for which the sum equals $\binom{k}{2}$.