Challenge: 2D Submatrix Count

Count submatrices summing to target $k$. This extends 1D subarray sum counting to 2D. Fix top and bottom rows. Compress columns into 1D array (column sums). Now apply 1D hashmap technique. For each pair of rows $(r1, r2)$, the column sums form a 1D array. Count subarrays summing to $k$ using the hashmap method. Time: $O(n^2 \cdot m)$

Time complexity: $O(n^2 \cdot m)$ for an $n \times m$ matrix. There are $O(n^2)$ row pairs, and each 1D pass takes $O(m)$.

Space complexity: $O(n \cdot m)$ for the prefix sum array.