Given two integers N and M. Consider two matrix ANXM, BNXM. 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 PNXM 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 matrixint 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 Programint 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 matriximport 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 matrixdef 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 CodeN = 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 matrixusing 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 matrixfunction 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 matrixfunction 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 codelet N = 3, M = 3; document.write(trace(N, M));// This code is contributed by susmitakundugoaldanga</script> |
Output
30
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,
Sn = (n * (a1 + an))/2
So, n = R, a1 = 1, an = 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, an = 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 APint sn(int n, int an){ return (n * (1 + an)) / 2;}// Return the trace of sum of row-major matrix// and column-major matrixint 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 Programint 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 matriximport 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 APdef sn(n, an): return (n * (1 + an)) / 2;# Return the trace of sum# of row-major matrix# and column-major matrixdef 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 CodeN = 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 matrixusing 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 APfunction sn($n, $an){ return ($n * (1 + $an)) / 2;}// Return the trace of sum// of row-major matrix// and column-major matrixfunction trace($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
30
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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