Notice that picking a row or column is equivalent to removing it from future consideration. The game ends when all rows and columns are depleted of 1s.

Model this as a bipartite graph where rows and columns are nodes, and each 1 is an edge. Nullifying a row/column is like removing a vertex and all incident edges. The game reduces to which player takes the last vertex.

Since removing a vertex also removes its incident edges, the game is equivalent to playing on the set of rows and columns that contain at least one 1. Alice wins if the count of such rows is different from the count of such columns. More precisely, let r = number of rows with at least one 1, c = number of columns with at least one 1. If r xor c != 0, Alice wins.