A special binary string has equal numbers of 0s and 1s, and any prefix has at least as many 1s as 0s. You need to recursively make the lexicographically largest special string possible from a given special binary string.

You are given a matrix and a series of update operations. Each operation specifies a submatrix that should have all its elements incremented by one. Your goal is to perform all the updates and return the modified matrix.

Find the longest string that is a prefix of all strings in a given array. If no common prefix exists, return an empty string.