# Sum of matrix in which each element is absolute difference of its row and column numbers

• Difficulty Level : Medium
• Last Updated : 13 Dec, 2021

Given a positive integer n. Consider a matrix of n rows and n columns, in which each element contain absolute difference of its row number and numbers. The task is to calculate sum of each element of the matrix.
Examples :

```Input : n = 2
Output : 2
Matrix formed with n = 2 with given constraint:
0 1
1 0
Sum of matrix = 2.

Input : n = 3
Output : 8
Matrix formed with n = 3 with given constraint:
0 1 2
1 0 1
2 1 0
Sum of matrix = 8.```

Method 1 (Brute Force):
Simply construct a matrix of n rows and n columns and initialize each cell with absolute difference of its corresponding row number and column number. Now, find the sum of each cell.
Below is the implementation of above idea :

## C++

 `// C++ program to find sum of matrix in which each``// element is absolute difference of its corresponding``// row and column number row.``#include``using` `namespace` `std;` `// Return the sum of matrix in which each element``// is absolute difference of its corresponding row``// and column number row``int` `findSum(``int` `n)``{``    ``// Generate matrix``    ``int` `arr[n][n];``    ``for` `(``int` `i = 0; i < n; i++)``        ``for` `(``int` `j = 0; j < n; j++)``            ``arr[i][j] = ``abs``(i - j);` `    ``// Compute sum``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``for` `(``int` `j = 0; j < n; j++)``            ``sum += arr[i][j];` `    ``return` `sum;``}` `// Driven Program``int` `main()``{``    ``int` `n = 3;``    ``cout << findSum(n) << endl;``    ``return` `0;``}`

## Java

 `// Java program to find sum of matrix``// in which each element is absolute``// difference of its corresponding``// row and column number row.``import` `java.io.*;` `public` `class` `GFG {` `// Return the sum of matrix in which``// each element is absolute difference``// of its corresponding row and column``// number row``static` `int` `findSum(``int` `n)``{``    ` `    ``// Generate matrix``    ``int` `[][]arr = ``new` `int``[n][n];``    ``for` `(``int` `i = ``0``; i < n; i++)``        ``for` `(``int` `j = ``0``; j < n; j++)``            ``arr[i][j] = Math.abs(i - j);` `    ``// Compute sum``    ``int` `sum = ``0``;``    ``for` `(``int` `i = ``0``; i < n; i++)``        ``for` `(``int` `j = ``0``; j < n; j++)``            ``sum += arr[i][j];` `    ``return` `sum;``}` `    ``// Driver Code``    ``static` `public` `void` `main (String[] args)``    ``{``        ``int` `n = ``3``;``        ``System.out.println(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## Python3

 `# Python3 program to find sum of matrix``# in which each element is absolute``# difference of its corresponding``# row and column number row.` `# Return the sum of matrix in which each``# element is absolute difference of its``# corresponding row and column number row``def` `findSum(n):` `    ``# Generate matrix``    ``arr ``=` `[[``0` `for` `x ``in` `range``(n)]``              ``for` `y ``in` `range` `(n)]``    ``for` `i ``in` `range` `(n):``        ``for` `j ``in` `range` `(n):``            ``arr[i][j] ``=` `abs``(i ``-` `j)` `    ``# Compute sum``    ``sum` `=` `0``    ``for` `i ``in` `range` `(n):``        ``for` `j ``in` `range``(n):``            ``sum` `+``=` `arr[i][j]` `    ``return` `sum` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``n ``=` `3``    ``print` `(findSum(n))``    ` `# This code is contributed by ita_c`

## C#

 `// C# program to find sum of matrix``// in which each element is absolute``// difference of its corresponding``// row and column number row.``using` `System;` `public` `class` `GFG {` `// Return the sum of matrix in which``// each element is absolute difference``// of its corresponding row and column``// number row``static` `int` `findSum(``int` `n)``{``    ` `// Generate matrix``    ``int` `[,]arr = ``new` `int``[n, n];``    ``for` `(``int` `i = 0; i < n; i++)``        ``for` `(``int` `j = 0; j < n; j++)``            ``arr[i,j ] = Math.Abs(i - j);`` ` `    ``// Compute sum``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``for` `(``int` `j = 0; j < n; j++)``            ``sum += arr[i, j];`` ` `    ``return` `sum;``}` `    ``// Driver Code``    ``static` `public` `void` `Main(String[] args)``    ``{``        ``int` `n = 3;``        ``Console.WriteLine(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 ``

## Javascript

 ``

Output:

`8`

Method 2 (O(n)):
Consider n = 3, matrix formed will be:
0 1 2
1 0 1
2 1 0
Observe, the main diagonal is always 0 since all i are equal to j. The diagonal just above and just below will always be 1 because at each cell either i is 1 greater than j or j is 1 greater than i and so on.
Following the pattern we can see that the total sum of all the elements in the matrix will be, for each i from 0 to n, add i*(n-i)*2.
Below is the implementation of above idea :

## C++

 `// C++ program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``#include``using` `namespace` `std;` `// Return the sum of matrix in which each``// element is absolute difference of its``// corresponding row and column number row``int` `findSum(``int` `n)``{``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``sum += i*(n-i);``    ``return` `2*sum;``}` `// Driven Program``int` `main()``{``    ``int` `n = 3;``    ``cout << findSum(n) << endl;``    ``return` `0;``}`

## Java

 `// Java program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``import` `java.io.*;` `class` `GFG {` `// Return the sum of matrix in which each``// element is absolute difference of its``// corresponding row and column number row``static` `int` `findSum(``int` `n)``{``    ``int` `sum = ``0``;``    ``for` `(``int` `i = ``0``; i < n; i++)``        ``sum += i * (n - i);``    ``return` `2` `* sum;``}` `    ``// Driver Code``    ``static` `public` `void` `main(String[] args)``    ``{``        ``int` `n = ``3``;``        ``System.out.println(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## C#

 `// C# program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``using` `System;` `class` `GFG {` `// Return the sum of matrix in which each``// element is absolute difference of its``// corresponding row and column number row``static` `int` `findSum(``int` `n)``{``    ``int` `sum = 0;``    ``for` `(``int` `i = 0; i < n; i++)``        ``sum += i * (n - i);``    ``return` `2 * sum;``}` `    ``// Driver Code``    ``static` `public` `void` `Main(String[] args)``    ``{``        ``int` `n = 3;``        ``Console.WriteLine(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## Python3

 `# Python 3 program to find sum``# of matrix in which each element``# is absolute difference of its``# corresponding row and column``# number row.` `# Return the sum of matrix in``# which each element is absolute``# difference of its corresponding``# row and column number row``def` `findSum(n):``    ``sum` `=` `0``    ``for` `i ``in` `range``(n):``        ``sum` `+``=` `i ``*` `(n ``-` `i)``    ``return` `2` `*` `sum` `# Driver code``n ``=` `3``print``(findSum(n))` `# This code is contributed by Shrikant13`

## PHP

 ``

## Javascript

 ``

Output:

`8`

Method 3 (Trick):
Consider n = 3, matrix formed will be:
0 1 2
1 0 1
2 1 0
So, sum = 1 + 1 + 1 + 1 + 2 + 2.
On Rearranging, 1 + 2 + 1 + 2 + 2 = 1 + 2 + 1 + 22
So, in every case we can rearrange the sum of matrix so that the answer always will be sum of first n – 1 natural number and sum of square of first n – 1 natural number.

```Sum of first n natural number = ((n)*(n + 1))/2.
Sum of first n natural number = ((n)*(n + 1)*(2*n + 1)/6.```

Below is the implementation of above idea :

## C++

 `// C++ program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``#include``using` `namespace` `std;` `// Return the sum of matrix in which each element``// is absolute difference of its corresponding``// row and column number row``int` `findSum(``int` `n)``{``    ``n--;``    ``int` `sum = 0;``    ``sum += (n*(n+1))/2;``    ``sum += (n*(n+1)*(2*n + 1))/6;``    ``return` `sum;``}` `// Driven Program``int` `main()``{``    ``int` `n = 3;``    ``cout << findSum(n) << endl;``    ``return` `0;``}`

## Java

 `// Java program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``import` `java.io.*;` `public` `class` `GFG {``    ` `// Return the sum of matrix in which each element``// is absolute difference of its corresponding``// row and column number row``static` `int` `findSum(``int` `n)``{``    ``n--;``    ``int` `sum = ``0``;``    ``sum += (n * (n + ``1``)) / ``2``;``    ``sum += (n * (n + ``1``) * (``2` `* n + ``1``)) / ``6``;``    ``return` `sum;``}` `    ``// Driver Code``    ``static` `public` `void` `main (String[] args)``    ``{``        ``int` `n = ``3``;``        ``System.out.println(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## Python3

 `# Python 3 program to find sum of matrix``# in which each element is absolute``# difference of its corresponding row``# and column number row.` `# Return the sum of matrix in which``# each element is absolute difference``# of its corresponding row and column``# number row``def` `findSum(n):``    ``n ``-``=` `1``    ``sum` `=` `0``    ``sum` `+``=` `(n ``*` `(n ``+` `1``)) ``/` `2``    ``sum` `+``=` `(n ``*` `(n ``+` `1``) ``*` `(``2` `*` `n ``+` `1``)) ``/` `6``    ``return` `int``(``sum``)` `# Driver Code``n ``=` `3``print``(findSum(n))` `# This code contributed by Rajput-Ji`

## C#

 `// C# program to find sum of matrix in which``// each element is absolute difference of its``// corresponding row and column number row.``using` `System;` `public` `class` `GFG {``    ` `// Return the sum of matrix in which each element``// is absolute difference of its corresponding``// row and column number row``static` `int` `findSum(``int` `n)``{``    ``n--;``    ``int` `sum = 0;``    ``sum += (n * (n + 1)) / 2;``    ``sum += (n * (n + 1) * (2 * n + 1)) / 6;``    ``return` `sum;``}` `    ``// Driver Code``    ``static` `public` `void` `Main(String[] args)``    ``{``        ``int` `n = 3;``        ``Console.WriteLine(findSum(n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 ``

## Javascript

 ``

Output :

`8`

Source:
https://stackoverflow.com/questions/42043708/sum-of-matrix-in-which-each-element-is-absolute-difference-of-row-and-column
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