The sum of all submatrices is beneficial in various scenarios, such as solving problems in **competitive programming, data analysis, and certain algorithms in image processing or computer vision**.

Understanding this concept can help in optimizing algorithms and reducing computational complexity in such tasks.

Here’s a breakdown of why the sum of all submatrices might be important

- Competitive Programming and Algorithmic Challenges
- Data Analysis
- Image Processing
- Optimization in Matrix Operations

Let’s understand with an example.

## Problem Description

Given a 2D Matrix A of dimensions

, we need to return the sum of all possible submatrices.**N*N**

### Problem Constraints

`1 <= N <=30`

0 <= A[i][j] <= 10

### Input Format

`Single argument representing a 2-D array A of size N x N.`

### Output Format

`Return an integer denoting the sum of all possible submatrices in the given matrix.`

#### Example Input & Output

```
Input 1:
A = [ [1, 1]
[1, 1] ]
Input 2:
A = [ [1, 2]
[3, 4] ]
Output 1:
16
Output 2:
40
```

### Output

##### Sum of all Submatrices in JAVA

```
public class Solution {
public int solve(int[][] A) {
int n = A.length;
int totalSum = 0;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
totalSum += A[i][j] * (i + 1) * (j + 1) * (n - i) * (n - j);
}
}
return totalSum;
}
}
```

##### Sum of all Submatrices in Python

```
def solve(A):
n = len(A)
total_sum = 0
for i in range(n):
for j in range(n):
total_sum += A[i][j] * (i + 1) * (j + 1) * (n - i) * (n - j)
return total_sum
```

##### Sum of all Submatrices in JavaScript

```
function solve(A) {
const n = A.length;
let totalSum = 0;
for (let i = 0; i < n; i++) {
for (let j = 0; j < n; j++) {
totalSum += A[i][j] * (i + 1) * (j + 1) * (n - i) * (n - j);
}
}
return totalSum;
}
```