# C Program To Check whether Matrix is Skew Symmetric or not?

Square Matrix A is said to be skew-symmetric if aij=−aji for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix A i.e (AT=−A).

Note that all the main diagonal elements in the skew-symmetric matrix are zero.

lets take an example of a matrix

A= |0 -5 4|
|5 0 -1|
|-4 1 0|

It is skew-symmetric matrix because aij=−aji for all i and j. Example, a12 = -5 and a21=5 which means a12=−a21. Similarly, this condition holds true for all other values of i and j.

We can also verify that Transpose of Matrix A is equal to negative of matrix A i.e AT=−A.

AT= |0 5 -4|
|-5 0 1|
|4 -1 0|
and
A= |0 -5 4|
|5 0 -1|
|-4 1 0|

We can clearly see that AT=−A which makes A skew-symmetric matrix.

Input:
Enter the number of rows and columns: 2 2
Enter the matrix elements: 10 20 20 10
Output:
The matrix is symmetric.
10 20
20 10

## Explanation

If the matrix is equal to its transpose, then it’s a symmetric matrix.

Else if it’s transpose is equal to the negative of itself, then the matrix is skew-symmetric. Else it is neither. The result is printed accordingly

The process to check for symmetry of a matrix

• The user is asked to enter a number of rows and columns of the matrix.

• The elements of the matrix are asked to enter and store in ‘A’. Variables ‘x’ and ‘y’ are initialized as 0.

• If the matrix is not equal to its transpose, a temporary variable ‘x’ is assigned 1.

• Else if the negative of the matrix is equal to its transpose, a temporary variable ‘y’ is assigned 1.

• If x is equal to 0, then the matrix is symmetric. Else if y is equal to 1, the matrix is skew-symmetric.

• If neither of the conditions satisfies, the matrix is neither symmetric nor skew-symmetric.

• The result is then printed.

## Example

#include<iostream>
using namespace std;
int main () {
int A[10][10], i, j, m, n, x = 0, y = 0;
cout << "Enter the number of rows and columns : ";
cin >> m >> n;
cout << "Enter the matrix elements : ";
for (i = 0; i < m; i++)
for (j = 0; j < n; j++)
cin >> A[i][j];
for (i = 0; i < m; i++) {
for( j = 0; j < n; j++) {
if (A[i][j] != A[j][i])
x = 1;
else if (A[i][j] == -A[j][i])
y = 1;
}
}
if (x == 0)
cout << "The matrix is symmetric. ";
else if (y == 1)
cout << "The matrix is skew symmetric. ";
else
cout << "It is neither symmetric nor skew-symmetric. ";
for (i = 0; i < m; i++) {
for (j = 0; j < n; j++)
cout << A[i][j] << " ";
cout << " ";
}
return 0;
}