# Matrix Block Sum

For each cell in the result array, we need to add some of the matrix elements based on the k value and row and column values. Easy and simple solution is given below:

This simple solution has a time complexity of O(m*n*k*k)

When computing the sums for certain elements in the resulting matrix `res`, we observe redundancy in the addition of elements. For instance, when calculating `res[0][0]`, elements `mat[0][0]`, `mat[0][1]`, `mat[1][0]`, and `mat[1][1]` are added. Then, when computing `res[0][1]`, these same elements are added again along with `mat[0][2]` and `mat[1][2]`, resulting in redundant computations. By trying to avoid these redundant computations, we can reduce time complexity.

Prefix sum

We can optimize the solution using the concept of prefix sum. Prefix sum is a technique used to compute the sum of elements in a sequence.

For a 1D array, we initialize `prefix_sum[0]` to 0, and then compute subsequent elements as `prefix_sum[i] = prefix_sum[i-1] + mat[i-1]`. This allows us to quickly calculate the sum of elements within a specified range from `i-k` to `i+k` using `prefix_sum[i+k+1] - prefix_sum[i-k+1]`.

Similarly, we can extend this concept to a 2D array to reduce the time complexity of the solution. By precomputing prefix sums along rows and columns, we can efficiently compute the sum of elements within a specified range for each cell in the resulting matrix.

Here is the solution:

This solution uses the prefix sum and dynamic programming concepts to reduce the time complexity.