Vasya and matrix

Vasya received a rectangular matrix of size n \times m as a gift from his mother. Each cell of the matrix contains an integer. At first, Vasya enthusiastically played various mathematical games with the matrix: sometimes he would swiftly compute its determinant, and sometimes he would easily raise it to different powers.

However, he eventually got bored of such games and came up with a new pastime: Vasya chooses an integer $k$ and tries to find a submatrix of maximum area such that the sum of all its elements does not exceed $k$. A submatrix is defined as a rectangular fragment of the original matrix.