Matrix: Given a matrix, check if it is orthogonal or not

Problem Statement:

You are given a matrix, you need to check if it is orthogonal or not.

A matrix is considered as orthogonal, when we multiply the orthogonal to its transpose, we get the identity matrix.

Example:

Input:

1 0 0
0 1 0 
0 0 1

Output:

Yes

Solution Explanation:

Solution is very simple.

First find the transpose of matrix.

Then multiply the transpose with the given matrix.

Then check if the matrix is identity or not.

Time Complexity: O(n*n*n)
Space Complexity: O(n*n)

Code Solution

#include <iostream>
#include <vector>
using namespace std;

const int MAX = 100;


bool solution(int a[][MAX], int m, int n) 
{
	if (m != n)
	    return false;


	int prod[n][n];

	for (int i = 0; i < n; i++) 
	{
	    for (int j = 0; j < n; j++) 
	    {

	    int sum = 0;
	    for (int k = 0; k < n; k++) 
	    {

	        sum = sum + (a[i][k] * a[j][k]);
	    }

	    prod[i][j] = sum;
	    }
	}

	for (int i = 0; i < n; i++) 
	{
	    for (int j = 0; j < n; j++) 
	    {
	    if (i != j && prod[i][j] != 0)
	        return false;
	    if (i == j && prod[i][j] != 1)
	        return false;
	    }
	}

	return true;
}

int main() 
{

int mat[][MAX] = {{1, 0, 0},
                {0, 1, 0},
                {0, 0, 1}};

if (solution(mat, 3, 3))
    cout << "Yes";
else
    cout << "No";
return 0;
}

Output

Yes