# Sum of all Submatrices of a Given Matrix

Given a **NxN** **2-D** matrix, the task to find the sum of all the submatrices.

**Examples:**

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Input :arr[] = {{1, 1}, {1, 1}};Output :16Explanation: Number of sub-matrices with 1 elements = 4 Number of sub-matrices with 2 elements = 4 Number of sub-matrices with 3 elements = 0 Number of sub-matrices with 4 elements = 1 Since all the entries are 1, the sum becomes sum = 1 * 4 + 2 * 4 + 3 * 0 + 4 * 1 = 16Input :arr[] = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}Output :500

**Simple Solution:** A naive solution is to generate all the possible submatrices and sum up all of them.

The time complexity of this approach will be O(n^{6}).

**Efficient Solution :** For each element of the matrix, let us try to find the number of sub-matrices, the element will lie in.

This can be done in O(1) time. Let us suppose the index of an element be (X, Y) in 0 based indexing, then the number of submatrices (S_{x, y}) for this element will be in can be given by the formula **S _{x, y} = (X + 1) * (Y + 1) * (N – X) * (N – Y)**. This formula works, because we just have to choose two different positions on the matrix that will create a submatrix that envelopes the element. Thus, for each element, ‘sum’ can be updated as

**sum += (S**.

_{x, y}) * Arr_{x, y}Below is the implementation of the above approach:

Here we need to try to solve this question in the **Reverse lookup Technique**:

1) For a particular element what are the **possible submatrices where this element will be included**.

2) When we get the number of possible submatrices then **we can count the contribution of that particular element by doing ( a[i]* total number of submatrices where this will be included) where a[i] = current element**

3) Now Question comes how to find the number of possible submatrices for a particular element.

[[1 2 3]

[4 5 6]

[7 8 9]]

So let’s consider the current element as 5 , so for 5 there are (X+1)*(Y+1) choices where co-ordinates of submatrix starting point can lie,(Top Left)

Similarly, there will be (N-X)*(N-Y) choices where the end co-ordinates of that submatrix can lie (Bottom Right)

** Number of choices for Top Left = (X+1)*(Y+1)**

**Number of choices for Bottom Right = (N-X)*(N-Y)**

Total number of choices for the current element to be included in the submatrix will be : (X+1)*(Y+1) * (N-X)*(N-Y)

**Contribution of the current element which can be included in all the possible submatrices will be = arr[X][Y] * (X+1)*(Y+1) * (N-X)*(N-Y)**

where **X and Y are index of the submatrices.**

**Time Complexity: O(N^2)**

**Space Complexity: O(1)**

## C++

`// C++ program to find the sum of all` `// possible submatrices of a given Matrix` `#include <iostream>` `#define n 3` `using` `namespace` `std;` `// Function to find the sum of all` `// possible submatrices of a given Matrix` `int` `matrixSum(` `int` `arr[][n])` `{` ` ` `// Variable to store` ` ` `// the required sum` ` ` `int` `sum = 0;` ` ` `// Nested loop to find the number` ` ` `// of submatrices, each number belongs to` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `for` `(` `int` `j = 0; j < n; j++) {` ` ` `// Number of ways to choose` ` ` `// from top-left elements` ` ` `int` `top_left = (i + 1) * (j + 1);` ` ` `// Number of ways to choose` ` ` `// from bottom-right elements` ` ` `int` `bottom_right = (n - i) * (n - j);` ` ` `sum += (top_left * bottom_right * arr[i][j]);` ` ` `}` ` ` `return` `sum;` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `arr[][n] = { { 1, 1, 1 },` ` ` `{ 1, 1, 1 },` ` ` `{ 1, 1, 1 } };` ` ` `cout << matrixSum(arr);` ` ` `return` `0;` `}` |

## Java

`// Java program to find the sum of all` `// possible submatrices of a given Matrix` `class` `GFG` `{` ` ` `static` `final` `int` `n = ` `3` `;` ` ` `// Function to find the sum of all` ` ` `// possible submatrices of a given Matrix` ` ` `static` `int` `matrixSum(` `int` `arr[][])` ` ` `{` ` ` `// Varialbe to store` ` ` `// the required sum` ` ` `int` `sum = ` `0` `;` ` ` `// Nested loop to find the number` ` ` `// of submatrices, each number belongs to` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++)` ` ` `{` ` ` `for` `(` `int` `j = ` `0` `; j < n; j++)` ` ` `{` ` ` `// Number of ways to choose` ` ` `// from top-left elements` ` ` `int` `top_left = (i + ` `1` `) * (j + ` `1` `);` ` ` `// Number of ways to choose` ` ` `// from bottom-right elements` ` ` `int` `bottom_right = (n - i) * (n - j);` ` ` `sum += (top_left * bottom_right * arr[i][j]);` ` ` `}` ` ` `}` ` ` `return` `sum;` ` ` `}` ` ` `// Driver Code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `int` `arr[][] = {{` `1` `, ` `1` `, ` `1` `},` ` ` `{` `1` `, ` `1` `, ` `1` `},` ` ` `{` `1` `, ` `1` `, ` `1` `}};` ` ` `System.out.println(matrixSum(arr));` ` ` `}` `}` `// This code contributed by Rajput-Ji` |

## Python3

`# Python3 program to find the sum of all` `# possible submatrices of a given Matrix` `n ` `=` `3` `# Function to find the sum of all` `# possible submatrices of a given Matrix` `def` `matrixSum(arr) :` ` ` ` ` `# Variable to store the required sum` ` ` `sum` `=` `0` `;` ` ` `# Nested loop to find the number of` ` ` `# submatrices, each number belongs to` ` ` `for` `i ` `in` `range` `(n) :` ` ` `for` `j ` `in` `range` `(n) :` ` ` `# Number of ways to choose` ` ` `# from top-left elements` ` ` `top_left ` `=` `(i ` `+` `1` `) ` `*` `(j ` `+` `1` `);` ` ` `# Number of ways to choose` ` ` `# from bottom-right elements` ` ` `bottom_right ` `=` `(n ` `-` `i) ` `*` `(n ` `-` `j);` ` ` `sum` `+` `=` `(top_left ` `*` `bottom_right ` `*` ` ` `arr[i][j]);` ` ` `return` `sum` `;` `# Driver Code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `arr ` `=` `[[ ` `1` `, ` `1` `, ` `1` `],` ` ` `[ ` `1` `, ` `1` `, ` `1` `],` ` ` `[ ` `1` `, ` `1` `, ` `1` `]];` ` ` `print` `(matrixSum(arr))` ` ` `# This code is contributed by Ryuga` |

## C#

`// C# program to find the sum of all` `// possible submatrices of a given Matrix` `using` `System;` `class` `GFG` `{` `static` `int` `n = 3;` `// Function to find the sum of all` `// possible submatrices of a given Matrix` `static` `int` `matrixSum(` `int` `[,]arr)` `{` ` ` `// Varialbe to store the` ` ` `// required sum` ` ` `int` `sum = 0;` ` ` `// Nested loop to find the number of ` ` ` `// submatrices, each number belongs to` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `{` ` ` `for` `(` `int` `j = 0; j < n; j++)` ` ` `{` ` ` `// Number of ways to choose` ` ` `// from top-left elements` ` ` `int` `top_left = (i + 1) * (j + 1);` ` ` `// Number of ways to choose` ` ` `// from bottom-right elements` ` ` `int` `bottom_right = (n - i) * (n - j);` ` ` `sum += (top_left * bottom_right * arr[i, j]);` ` ` `}` ` ` `}` ` ` `return` `sum;` `}` `// Driver Code` `public` `static` `void` `Main()` `{` ` ` `int` `[,]arr = {{1, 1, 1},` ` ` `{1, 1, 1},` ` ` `{1, 1, 1}};` ` ` `Console.WriteLine(matrixSum(arr));` `}` `}` `// This code contributed by vt_m..` |

## PHP

`<?php` `// PHP program to find the sum of all` `// possible submatrices of a given Matrix` `// Function to find the sum of all` `// possible submatrices of a given Matrix` `function` `matrixSum(` `$arr` `)` `{` ` ` `$n` `= 3;` ` ` ` ` `// Variable to store the required sum` ` ` `$sum` `= 0;` ` ` `// Nested loop to find the number` ` ` `// of submatrices, each number belongs to` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++)` ` ` `for` `(` `$j` `= 0; ` `$j` `< ` `$n` `; ` `$j` `++)` ` ` `{` ` ` `// Number of ways to choose` ` ` `// from top-left elements` ` ` `$top_left` `= (` `$i` `+ 1) * (` `$j` `+ 1);` ` ` `// Number of ways to choose` ` ` `// from bottom-right elements` ` ` `$bottom_right` `= (` `$n` `- ` `$i` `) * (` `$n` `- ` `$j` `);` ` ` `$sum` `+= (` `$top_left` `* ` `$bottom_right` `*` ` ` `$arr` `[` `$i` `][` `$j` `]);` ` ` `}` ` ` `return` `$sum` `;` `}` `// Driver Code` `$arr` `= ` `array` `(` `array` `(1, 1, 1),` ` ` `array` `(1, 1, 1),` ` ` `array` `(1, 1, 1));` `echo` `matrixSum(` `$arr` `);` `// This code is contributed` `// by Akanksha Rai` `?>` |

## Javascript

`<script>` `// JavaScript program to find the sum of all` `// possible submatrices of a given Matrix` `let n = 3;` `// Function to find the sum of all` `// possible submatrices of a given Matrix` `function` `matrixSum(arr)` `{` ` ` `// Variable to store` ` ` `// the required sum` ` ` `let sum = 0;` ` ` `// Nested loop to find the number` ` ` `// of submatrices, each number belongs to` ` ` `for` `(let i = 0; i < n; i++)` ` ` `for` `(let j = 0; j < n; j++) {` ` ` `// Number of ways to choose` ` ` `// from top-left elements` ` ` `let top_left = (i + 1) * (j + 1);` ` ` `// Number of ways to choose` ` ` `// from bottom-right elements` ` ` `let bottom_right = (n - i) * (n - j);` ` ` `sum += (top_left * bottom_right * arr[i][j]);` ` ` `}` ` ` `return` `sum;` `}` `// Driver Code` `let arr = [[ 1, 1, 1 ],` ` ` `[ 1, 1, 1 ],` ` ` `[ 1, 1, 1 ]] ;` ` ` `document.write(matrixSum(arr));` ` ` `// This code is contributed by todaysgaurav` `</script>` |

**Output:**

100