Find trace of matrix formed by adding Row-major and Column-major order of same matrix

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Given two integers N and M. Consider two matrix A_NXM, B_NXM. Both matrix A and matrix B contains elements from 1 to N*M. Matrix A contains elements in Row-major order and matrix B contains elements in Column-major order. The task is to find the trace of the matrix formed by addition of A and B. Trace of matrix P_NXM is defined as P[0][0] + P[1][1] + P[2][2] +….. + P[min(n – 1, m – 1)][min(n – 1, m – 1)] i.e addition of main diagonal.

Note – Both matrix A and matrix B contain elements from 1 to N*M.

Examples :

Input : N = 3, M = 3
Output : 30
Therefore,
    1 2 3
A = 4 5 6
    7 8 9

    1 4 7
B = 2 5 8
    3 6 9
  
        2 6 10
A + B = 6 10 14
       10 14 18

Trace = 2 + 10 + 18 = 30

Method 1 (Naive Approach): Generate matrix A and B and find the sum. Then traverse the main diagonal and find the sum.

Below is the implementation of this approach:

C++

// C++ program to find
// trace of matrix formed by
// adding Row-major and
// Column-major order of same matrix
#include <bits/stdc++.h>
using namespace std;
 
// Return the trace of
// sum of row-major matrix
// and column-major matrix
int trace(int n, int m)
{
 
    int A[n][m], B[n][m], C[n][m];    
 
    // Generating the matrix A
    int cnt = 1;
    for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++) {
            A[i][j] = cnt;
            cnt++;
        }    
 
    // Generating the matrix A
    cnt = 1;
    for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++) {
            B[j][i] = cnt;
            cnt++;
        }
 
    // Finding sum of matrix A and matrix B
    for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++)
            C[i][j] = A[i][j] + B[i][j];    
 
    // Finding the trace of matrix C.
    int sum = 0;
    for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++)
            if (i == j)
                sum += C[i][j];
 
    return sum;
}
 
// Driven Program
int main()
{
    int N = 3, M = 3;
    cout << trace(N, M) << endl;
    return 0;
}

Java

// Java program to find
// trace of matrix formed by
// adding Row-major and
// Column-major order of same matrix
import java.io.*;
 
public class GFG
{
    // Return the trace of
    // sum of row-major matrix
    // and column-major matrix
    static int trace(int n, int m)
    {
     
        int A[][] = new int[n][m];
        int B[][] = new int[n][m];
        int C[][] = new int[n][m];
     
        // Generating the matrix A
        int cnt = 1;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                A[i][j] = cnt;
                cnt++;
            } 
     
        // Generating the matrix A
        cnt = 1;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                B[j][i] = cnt;
                cnt++;
            }
     
        // Finding sum of matrix A and matrix B
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                C[i][j] = A[i][j] + B[i][j]; 
     
        // Finding the trace of matrix C.
        int sum = 0;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                if (i == j)
                    sum += C[i][j];
     
        return sum;
    }
     
    // Driver code 
    public static void main (String[] args)
    {
        int N = 3, M = 3;
         
        System.out.println(trace(N, M));
    }
}
 
// This code is contributed by Anant Agarwal.

Python3

# Python3 program to find trace of matrix 
# formed by adding Row-major and
# Column-major order of same matrix
 
# Return the trace of sum of row-major 
# matrix and column-major matrix
def trace(n, m):
 
    A = [[0 for x in range(m)] 
            for y in range(n)];
    B = [[0 for x in range(m)] 
            for y in range(n)];
    C = [[0 for x in range(m)] 
            for y in range(n)]; 
 
    # Generating the matrix A
    cnt = 1;
    for i in range(n):
        for j in range(m):
            A[i][j] = cnt;
            cnt += 1;
 
    # Generating the matrix A
    cnt = 1;
    for i in range(n):
        for j in range(m):
            B[j][i] = cnt;
            cnt += 1;
 
    # Finding sum of matrix A and matrix B
    for i in range(n):
        for j in range(m):
            C[i][j] = A[i][j] + B[i][j]; 
 
    # Finding the trace of matrix C.
    sum = 0;
    for i in range(n):
        for j in range(m):
            if (i == j):
                sum += C[i][j];
 
    return sum;
 
# Driver Code
N = 3; 
M = 3;
print(trace(N, M));
     
# This code is contributed by mits

C#

// C# program to find
// trace of matrix formed by
// adding Row-major and
// Column-major order of same matrix
using System;
 
class GFG {
     
    // Return the trace of
    // sum of row-major matrix
    // and column-major matrix
    static int trace(int n, int m)
    {
        int[, ] A = new int[n, m];
        int[, ] B = new int[n, m];
        int[, ] C = new int[n, m];
 
        // Generating the matrix A
        int cnt = 1;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                A[i, j] = cnt;
                cnt++;
            }
 
        // Generating the matrix A
        cnt = 1;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                B[j, i] = cnt;
                cnt++;
            }
 
        // Finding sum of matrix A and matrix B
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                C[i, j] = A[i, j] + B[i, j];
 
        // Finding the trace of matrix C.
        int sum = 0;
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                if (i == j)
                    sum += C[i, j];
 
        return sum;
    }
 
    // Driver code
    public static void Main()
    {
        int N = 3, M = 3;
        Console.WriteLine(trace(N, M));
    }
}
 
// This code is contributed by vt_m.

PHP

<?php
// PHP program to find trace of matrix 
// formed by adding Row-major and
// Column-major order of same matrix
 
// Return the trace of sum of row-major 
// matrix and column-major matrix
function trace($n, $m)
{
 
    $A = array_fill(0, $n, array_fill(0, $m, 0));
    $B = array_fill(0, $n, array_fill(0, $m, 0));
    $C = array_fill(0, $n, array_fill(0, $m, 0)); 
 
    // Generating the matrix A
    $cnt = 1;
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $m; $j++)
        {
            $A[$i][$j] = $cnt;
            $cnt++;
        } 
 
    // Generating the matrix A
    $cnt = 1;
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $m; $j++)
        {
            $B[$j][$i] = $cnt;
            $cnt++;
        }
 
    // Finding sum of matrix A and matrix B
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $m; $j++)
            $C[$i][$j] = $A[$i][$j] + $B[$i][$j]; 
 
    // Finding the trace of matrix C.
    $sum = 0;
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $m; $j++)
            if ($i == $j)
                $sum += $C[$i][$j];
 
    return $sum;
}
 
// Driver Code
$N = 3; 
$M = 3;
print(trace($N, $M));
     
// This code is contributed by mits
?>

Javascript

<script>
 
// Javascript program to find
// trace of matrix formed by
// adding Row-major and
// Column-major order of same matrix
 
// Return the trace of
// sum of row-major matrix
// and column-major matrix
function trace(n, m)
{
   
    let A = new Array(n);
     
    // Loop to create 2D array using 1D array
    for(var i = 0; i < A.length; i++) 
    {
        A[i] = new Array(2);
    }
     
    let B = new Array(n);
     
    // Loop to create 2D array using 1D array
    for(var i = 0; i < B.length; i++) 
    {
        B[i] = new Array(2);
    }
     
    let C = new Array(n);
     
    // Loop to create 2D array using 1D array
    for(var i = 0; i < C.length; i++)
    {
        C[i] = new Array(2);
    }
     
    // Generating the matrix A
    let cnt = 1;
    for(let i = 0; i < n; i++)
        for(let j = 0; j < m; j++) 
        {
            A[i][j] = cnt;
            cnt++;
        } 
     
    // Generating the matrix A
    cnt = 1;
    for(let i = 0; i < n; i++)
        for(let j = 0; j < m; j++) 
        {
            B[j][i] = cnt;
            cnt++;
        }
     
    // Finding sum of matrix A and matrix B
    for(let i = 0; i < n; i++)
        for(let j = 0; j < m; j++)
            C[i][j] = A[i][j] + B[i][j]; 
     
    // Finding the trace of matrix C.
    let sum = 0;
    for(let i = 0; i < n; i++)
        for(let j = 0; j < m; j++)
            if (i == j)
                sum += C[i][j];
     
    return sum;
} 
 
// Driver code
let N = 3, M = 3;
   
document.write(trace(N, M));
 
// This code is contributed by susmitakundugoaldanga
 
</script>

Output

Time Complexity: O(N*M), as we are traversing the matrix using nested loops.
Auxiliary Space: O(N*M), as we are using extra space.

Method 2 (efficient approach) :

Basically, we need to find the sum of main diagonal of the first matrix A and main diagonal of the second matrix B.
Let’s take an example, N = 3, M = 4.
Therefore, Row-major matrix will be,

     1  2  3  4
A =  5  6  7  8
     9 10 11 12

So, we need the sum of 1, 6, 11.
Observe, it forms an Arithmetic Progression with a constant difference of a number of columns, M.
Also, first element is always 1. So, AP formed in case of Row-major matrix is 1, 1+(M+1), 1+2*(M+1), ….. consisting of N (number of rows) elements. And we know,
S_n = (n * (a₁ + a_n))/2
So, n = R, a₁ = 1, a_n = 1 + (R – 1)*(M+1).
Similarly, in case of Column-major, AP formed will be 1, 1+(N+1), 1+2*(N+1), …..
So, n = R, a₁ = 1, a_n = 1 + (R – 1)*(N+1).

Below is the implementation of this approach:

C++

// C++ program to find trace of matrix formed
// by adding Row-major and Column-major order
// of same matrix
#include <bits/stdc++.h>
using namespace std;
 
// Return sum of first n integers of an AP
int sn(int n, int an)
{
    return (n * (1 + an)) / 2;
}
 
// Return the trace of sum of row-major matrix
// and column-major matrix
int trace(int n, int m)
{
    // Finding nth element in
    // AP in case of Row major matrix.
    int an = 1 + (n - 1) * (m + 1);
 
    // Finding sum of first n integers
    // of AP in case of Row major matrix
    int rowmajorSum = sn(n, an);
 
    // Finding nth element in AP
    // in case of Row major matrix
    an = 1 + (n - 1) * (n + 1);
 
    // Finding sum of first n integers
    // of AP in case of Column major matrix
    int colmajorSum = sn(n, an);
 
    return rowmajorSum + colmajorSum;
}
 
// Driven Program
int main()
{
    int N = 3, M = 3;
    cout << trace(N, M) << endl;
    return 0;
}

Java

// Java program to find trace of matrix formed
// by adding Row-major and Column-major order
// of same matrix
import java.io.*;
 
public class GFG {
 
    // Return sum of first n integers of an AP
    static int sn(int n, int an)
    {
        return (n * (1 + an)) / 2;
    }
 
    // Return the trace of sum of row-major matrix
    // and column-major matrix
    static int trace(int n, int m)
    {
        // Finding nth element in
        // AP in case of Row major matrix.
        int an = 1 + (n - 1) * (m + 1);
 
        // Finding sum of first n integers
        // of AP in case of Row major matrix
        int rowmajorSum = sn(n, an);
 
        // Finding nth element in AP
        // in case of Row major matrix
        an = 1 + (n - 1) * (n + 1);
 
        // Finding sum of first n integers
        // of AP in case of Column major matrix
        int colmajorSum = sn(n, an);
 
        return rowmajorSum + colmajorSum;
    }
 
    // Driven Program
    static public void main(String[] args)
    {
        int N = 3, M = 3;
        System.out.println(trace(N, M));
    }
}
 
// This code is contributed by vt_m.

Python3

# Python3 program to find trace 
# of matrix formed by adding
# Row-major and Column-major 
# order of same matrix
 
# Return sum of first n 
# integers of an AP
def sn(n, an):
    return (n * (1 + an)) / 2;
 
# Return the trace of sum
# of row-major matrix
# and column-major matrix
def trace(n, m):
     
    # Finding nth element 
    # in AP in case of
    # Row major matrix.
    an = 1 + (n - 1) * (m + 1);
     
    # Finding sum of first
    # n integers of AP in
    # case of Row major matrix
    rowmajorSum = sn(n, an);
     
    # Finding nth element in AP
    # in case of Row major matrix
    an = 1 + (n - 1) * (n + 1);
     
    # Finding sum of first n 
    # integers of AP in case 
    # of Column major matrix
    colmajorSum = sn(n, an);
     
    return int(rowmajorSum +
               colmajorSum);
     
# Driver Code
N = 3;
M = 3;
print(trace(N, M));
 
# This code is contributed mits

C#

// C# program to find trace of matrix formed
// by adding Row-major and Column-major order
// of same matrix
using System;
 
public class GFG {
 
    // Return sum of first n integers of an AP
    static int sn(int n, int an)
    {
        return (n * (1 + an)) / 2;
    }
 
    // Return the trace of sum of row-major matrix
    // and column-major matrix
    static int trace(int n, int m)
    {
        // Finding nth element in
        // AP in case of Row major matrix.
        int an = 1 + (n - 1) * (m + 1);
 
        // Finding sum of first n integers
        // of AP in case of Row major matrix
        int rowmajorSum = sn(n, an);
 
        // Finding nth element in AP
        // in case of Row major matrix
        an = 1 + (n - 1) * (n + 1);
 
        // Finding sum of first n integers
        // of AP in case of Column major matrix
        int colmajorSum = sn(n, an);
 
        return rowmajorSum + colmajorSum;
    }
 
    // Driven Program
    static public void Main()
    {
        int N = 3, M = 3;
        Console.WriteLine(trace(N, M));
    }
}
 
// This code is contributed by vt_m.

PHP

<?php
// PHP program to find trace of matrix formed
// by adding Row-major and Column-major order
// of same matrix
 
// Return sum of first n integers of an AP
function sn($n, $an)
{
    return ($n * (1 + $an)) / 2;
}
 
// Return the trace of sum
// of row-major matrix
// and column-major matrix
function trace($n, $m)
{
     
    // Finding nth element in
    // AP in case of Row major matrix.
    $an = 1 + ($n - 1) * ($m + 1);
 
    // Finding sum of first n integers
    // of AP in case of Row major matrix
    $rowmajorSum = sn($n, $an);
 
    // Finding nth element in AP
    // in case of Row major matrix
    $an = 1 + ($n - 1) * ($n + 1);
 
    // Finding sum of first n integers
    // of AP in case of Column major matrix
    $colmajorSum = sn($n, $an);
 
    return $rowmajorSum + $colmajorSum;
}
     
    // Driver Code
    $N = 3;
    $M = 3;
    echo trace($N, $M),"\n";
 
// This code is contributed ajit
?>

Javascript

<script>
 
// Javascript program to find trace of matrix formed
// by adding Row-major and Column-major order
// of same matrix
 
    // Return sum of first n integers of an AP
    function sn(n,an)
    {
        return (n * (1 + an)) / 2;
    }
 
    // Return the trace of sum of row-major matrix
    // and column-major matrix
    function trace(n,m)
    {
        // Finding nth element in
        // AP in case of Row major matrix.
        let an = 1 + (n - 1) * (m + 1);
 
        // Finding sum of first n integers
        // of AP in case of Row major matrix
        let rowmajorSum = sn(n, an);
 
        // Finding nth element in AP
        // in case of Row major matrix
        an = 1 + (n - 1) * (n + 1);
 
        // Finding sum of first n integers
        // of AP in case of Column major matrix
        let colmajorSum = sn(n, an);
 
        return rowmajorSum + colmajorSum;
    }
 
    // Driven Program
     
        let N = 3, M = 3;
        document.write(trace(N, M));
 
// This code is contributed
// by sravan kumar Gottumukkala
 
</script>

Output

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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