Sum of matrix in which each element is absolute difference of its row and column numbers
Last Updated :
12 Sep, 2023
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;
// 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 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 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# 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
// 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)
{
// 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.
?>
<script>
// Javascript 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)
{
// Generate matrix
let arr=new Array(n);
for(let i = 0; i < n; i++)
{
arr[i] = new Array(n);
for(let j = 0; j < n; j++)
{
arr[i][j] = 0;
}
}
for (let i = 0; i < n; i++)
for (let j = 0; j < n; j++)
arr[i][j] = Math.abs(i - j);
// Compute sum
let sum = 0;
for (let i = 0; i < n; i++)
for (let j = 0; j < n; j++)
sum += arr[i][j];
return sum;
}
// Driver Code
let n = 3;
document.write(findSum(n));
// This code is contributed by avanitrachhadiya2155
</script>
Time Complexity: O(N2), as we are traversing the matrix using nested loops.
Auxiliary Space: O(N2), as we are using extra space for generating and storing the Matrix.
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;
// 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 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# 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.
# 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
// 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.
?>
<script>
// Java script 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)
{
let sum = 0;
for (let i = 0; i < n; i++)
sum += i * (n - i);
return 2 * sum;
}
// Driver Code
let n = 3;
document.write(findSum(n));
// This code is contributed by mohan pavan
</script>
Time Complexity: O(N), as we are only using single loop to traverse.
Auxiliary Space: O(1), as we are not using any extra space.
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<bits/stdc++.h>
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 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.
# 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# 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
// 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)
{
$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.
?>
<script>
// Java script 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)
{
n--;
let sum = 0;
sum += (n * (n + 1)) / 2;
sum += (n * (n + 1) * (2 * n + 1)) / 6;
return sum;
}
// Driver Code
let n = 3;
document.write(findSum(n));
// This code is contributed by mohan pavan
</script>
Time Complexity: O(1), as we are not using any loops.
Auxiliary Space: O(1), as we are not using any extra space.
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