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<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Retuen 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; ` `} ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## 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 ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## PHP

`<?php ` `// PHP program to find sum of ` `// matrix in which each element ` `// is absolute difference of ` `// its corresponding row and ` `// column number row. ` ` ` `// Retuen the sum of matrix ` `// in which each element ` `// is absolute difference ` `// of its corresponding row ` `// and column number row ` `function` `findSum( ` `$n` `) ` `{ ` ` ` ` ` `// Generate matrix ` ` ` `$arr` `=` `array` `(` `array` `()); ` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++) ` ` ` `for` `(` `$j` `= 0; ` `$j` `< ` `$n` `; ` `$j` `++) ` ` ` `$arr` `[` `$i` `][` `$j` `] = ` `abs` `(` `$i` `- ` `$j` `); ` ` ` ` ` `// Compute sum ` ` ` `$sum` `= 0; ` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++) ` ` ` `for` `(` `$j` `= 0; ` `$j` `< ` `$n` `; ` `$j` `++) ` ` ` `$sum` `+= ` `$arr` `[` `$i` `][` `$j` `]; ` ` ` ` ` `return` `$sum` `; ` `} ` ` ` ` ` `// Driver Code ` ` ` `$n` `= 3; ` ` ` `echo` `findSum(` `$n` `); ` ` ` `// This code is contributed by anuj_67. ` `?> ` |

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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<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Retuen 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; ` `} ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## 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 ` |

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## PHP

`<?php ` `// PHP 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 ` `function` `findSum(` `$n` `) ` `{ ` ` ` `$sum` `= 0; ` ` ` `for` `( ` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++) ` ` ` `$sum` `+= ` `$i` `* (` `$n` `- ` `$i` `); ` ` ` `return` `2 * ` `$sum` `; ` `} ` ` ` ` ` `// Driver Code ` ` ` `$n` `= 3; ` ` ` `echo` `findSum(` `$n` `); ` ` ` `// This code is contributed by anuj_67. ` `?> ` |

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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 + 2^{2}.

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<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Retuen 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; ` `} ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## 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 ` |

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## 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 { ` ` ` `// Retuen 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. ` |

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## PHP

`<?php ` `// PHP program to find sum of ` `// matrix in which each element ` `// is absolute difference of its ` `// corresponding row and column ` `// number row. ` ` ` `// Retuen the sum of matrix in ` `// which each element is absolute ` `// difference of its corresponding ` `// row and column number row ` `function` `findSum(` `$n` `) ` `{ ` ` ` `$n` `--; ` ` ` `$sum` `= 0; ` ` ` `$sum` `+= (` `$n` `* (` `$n` `+ 1)) / 2; ` ` ` `$sum` `+= (` `$n` `* (` `$n` `+ 1) * ` ` ` `(2 * ` `$n` `+ 1)) / 6; ` ` ` `return` `$sum` `; ` `} ` ` ` `// Driver Code ` `$n` `= 3; ` `echo` `findSum(` `$n` `) ; ` ` ` `// This code is contributed ` `// by nitin mittal. ` `?> ` |

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**Output :**

8

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