A set of nonzero entries of a (0,1)-matrix is an isolated set if no two entries belong to the same row, no two entries belong to the same column, and no two entries belong to a submatrix of the form [1 1; 1 1]. The isolation number of a matrix is the maximum size over all isolated sets. The isolation number of a matrix is a well-known and well-used lower bound for the matrix's Boolean rank. We will discuss the isolation number of the adjacency matrix of various graphs and develop some extremal results for n x n matrices with isolation number n.