Check if Matrix is Symmetric in C: Program to Find Symmetric Matrix

📚 In this tutorial, you will learn:

What is a symmetric matrix and its properties
How to check if a matrix is symmetric
How to use the transpose condition for verification
Step-by-step explanation of the program
Practice exercises to test your understanding

Introduction

In this tutorial, we will learn how to write a C program to check if a matrix is symmetric.

A symmetric matrix is a square matrix that is equal to its transpose. In other words, A[i][j] = A[j][i] for all i and j. This means the matrix is symmetric about its main diagonal.

💡 Key Point: A symmetric matrix must be a square matrix (same number of rows and columns). Non-square matrices cannot be symmetric.

Symmetric matrices are important in many real-world applications, such as:

Linear Algebra: Eigenvalue problems, quadratic forms
Physics: Inertia tensors, stress tensors
Statistics: Covariance and correlation matrices
Computer Graphics: Rotation and transformation matrices

Visual Example

Symmetric Matrix (3×3):

[1  2  3]
[2  4  5]
[3  5  6]

Transpose of Symmetric Matrix:

[1  2  3]
[2  4  5]
[3  5  6]

✅ Matrix = Transpose → Symmetric

Check: A[0][1] = 2, A[1][0] = 2 ✓
A[0][2] = 3, A[2][0] = 3 ✓
A[1][2] = 5, A[2][1] = 5 ✓

❌ Non-Symmetric Matrix:

[1  2  3]
[4  5  6]
[7  8  9]

Check: A[0][1] = 2, A[1][0] = 4 → Not equal!

C Program to Check if Matrix is Symmetric

#include <stdio.h>

int main() {
    int n, i, j;
    int symmetric = 1;  // Flag: 1 = symmetric, 0 = not symmetric
    
    // Ask user for matrix size
    printf("Enter the size of the square matrix: ");
    scanf("%d", &n);
    
    // Declare matrix
    int matrix[n][n];
    
    // Read elements into the matrix
    printf("\nEnter %d elements:\n", n * n);
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            printf("matrix[%d][%d] = ", i, j);
            scanf("%d", &matrix[i][j]);
        }
    }
    
    // Display the matrix
    printf("\n=== The Matrix ===\n");
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            printf("%d\t", matrix[i][j]);
        }
        printf("\n");
    }
    
    // Check if matrix is symmetric
    // Only need to check upper triangle (i < j)
    for(i = 0; i < n; i++) {
        for(j = i + 1; j < n; j++) {
            if(matrix[i][j] != matrix[j][i]) {
                symmetric = 0;
                break;
            }
        }
        if(symmetric == 0) {
            break;
        }
    }
    
    // Display result
    printf("\n=== Result ===\n");
    if(symmetric == 1) {
        printf("✅ The matrix is SYMMETRIC.\n");
        printf("(Matrix is equal to its transpose)\n");
    } else {
        printf("❌ The matrix is NOT symmetric.\n");
        printf("(Matrix is not equal to its transpose)\n");
    }
    
    return 0;
}

Sample Output

Enter the size of the square matrix: 3

Enter 9 elements:
matrix[0][0] = 1
matrix[0][1] = 2
matrix[0][2] = 3
matrix[1][0] = 2
matrix[1][1] = 4
matrix[1][2] = 5
matrix[2][0] = 3
matrix[2][1] = 5
matrix[2][2] = 6

=== The Matrix ===
1	2	3
2	4	5
3	5	6

=== Result ===
✅ The matrix is SYMMETRIC.
(Matrix is equal to its transpose)

When Matrix is Not Symmetric:

Enter the size of the square matrix: 3

Enter 9 elements:
matrix[0][0] = 1
matrix[0][1] = 2
matrix[0][2] = 3
matrix[1][0] = 4
matrix[1][1] = 5
matrix[1][2] = 6
matrix[2][0] = 7
matrix[2][1] = 8
matrix[2][2] = 9

=== The Matrix ===
1	2	3
4	5	6
7	8	9

=== Result ===
❌ The matrix is NOT symmetric.
(Matrix is not equal to its transpose)

Example with Identity Matrix (Always Symmetric):

Enter the size of the square matrix: 4

Enter 16 elements:
matrix[0][0] = 1
matrix[0][1] = 0
matrix[0][2] = 0
matrix[0][3] = 0
matrix[1][0] = 0
matrix[1][1] = 1
matrix[1][2] = 0
matrix[1][3] = 0
matrix[2][0] = 0
matrix[2][1] = 0
matrix[2][2] = 1
matrix[2][3] = 0
matrix[3][0] = 0
matrix[3][1] = 0
matrix[3][2] = 0
matrix[3][3] = 1

=== The Matrix ===
1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

=== Result ===
✅ The matrix is SYMMETRIC.
(Matrix is equal to its transpose)

Program Explanation

Let's break down the code step by step:

Include Header File: #include <stdio.h> includes the standard input/output library.
Declare Variables:
- int n; — size of the square matrix
- int i, j; — loop counters
- int symmetric = 1; — flag variable (1 = symmetric, 0 = not symmetric)
Read Matrix: Reads n × n elements into the matrix.
Check Symmetry:
- Only the upper triangle (i < j) needs to be checked
- Diagonal elements (i == j) are always equal to themselves
- If matrix[i][j] != matrix[j][i], set symmetric = 0 and break
Display Result: Prints whether the matrix is symmetric or not.
Return: return 0; indicates successful program execution.

📝 Note: We only need to check the upper triangle because symmetry is a symmetric property. If A[i][j] = A[j][i] for all i < j, then it automatically holds for all i > j as well.

Algorithm to Check Symmetric Matrix

Step-by-step algorithm:

Start
Read the size of the square matrix (n)
Read n × n elements into the matrix
Set symmetric = 1
For i = 0 to n-1:
- For j = i+1 to n-1:
  - If matrix[i][j] != matrix[j][i]:
    - Set symmetric = 0
    - Break
- If symmetric == 0, break
If symmetric == 1:
- Print "Matrix is symmetric"
Else:
- Print "Matrix is not symmetric"
End

Alternative Method: Using Transpose

Another way to check if a matrix is symmetric is to find its transpose and compare it with the original matrix:

#include <stdio.h>

int main() {
    int n, i, j;
    int symmetric = 1;
    
    printf("Enter the size of the square matrix: ");
    scanf("%d", &n);
    
    int matrix[n][n];
    int transpose[n][n];
    
    printf("\nEnter %d elements:\n", n * n);
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            scanf("%d", &matrix[i][j]);
        }
    }
    
    // Find transpose
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            transpose[j][i] = matrix[i][j];
        }
    }
    
    // Compare matrix with its transpose
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            if(matrix[i][j] != transpose[i][j]) {
                symmetric = 0;
                break;
            }
        }
        if(symmetric == 0) {
            break;
        }
    }
    
    printf("\n=== Matrix ===\n");
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            printf("%d\t", matrix[i][j]);
        }
        printf("\n");
    }
    
    printf("\n=== Transpose ===\n");
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            printf("%d\t", transpose[i][j]);
        }
        printf("\n");
    }
    
    printf("\n=== Result ===\n");
    if(symmetric == 1) {
        printf("✅ Matrix is SYMMETRIC (Matrix = Transpose)\n");
    } else {
        printf("❌ Matrix is NOT symmetric (Matrix ≠ Transpose)\n");
    }
    
    return 0;
}

Properties of Symmetric Matrices

Property	Description
Square Matrix	Must have equal rows and columns
A = A^T	Matrix equals its transpose
Diagonal Elements	Any values (not required to be equal)
Off-Diagonal Elements	A[i][j] = A[j][i] for all i ≠ j
Eigenvalues	All eigenvalues are real
Eigenvectors	Eigenvectors are mutually orthogonal

Time and Space Complexity

Method	Time Complexity	Space Complexity
Upper Triangle Check	O(n²)	O(1)
Transpose Method	O(n²)	O(n²)

💻 Practice Exercise

Challenge 1: Write a program to check if a matrix is skew-symmetric (A[i][j] = -A[j][i]).

Challenge 2: Create a function that returns 1 if a matrix is symmetric and 0 otherwise.

🔍 Click to Show Solution for Challenge 1

#include <stdio.h>

int main() {
    int n, i, j;
    int skew_symmetric = 1;
    
    printf("Enter the size of the square matrix: ");
    scanf("%d", &n);
    
    int matrix[n][n];
    
    printf("\nEnter %d elements:\n", n * n);
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            scanf("%d", &matrix[i][j]);
        }
    }
    
    // Check if matrix is skew-symmetric
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            // Check condition: A[i][j] = -A[j][i]
            if(matrix[i][j] != -matrix[j][i]) {
                skew_symmetric = 0;
                break;
            }
        }
        if(skew_symmetric == 0) {
            break;
        }
    }
    
    printf("\n=== The Matrix ===\n");
    for(i = 0; i < n; i++) {
        for(j = 0; j < n; j++) {
            printf("%d\t", matrix[i][j]);
        }
        printf("\n");
    }
    
    printf("\n=== Result ===\n");
    if(skew_symmetric == 1) {
        printf("✅ The matrix is SKEW-SYMMETRIC.\n");
    } else {
        printf("❌ The matrix is NOT skew-symmetric.\n");
    }
    
    return 0;
}

Frequently Asked Questions

1. What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose. This means A[i][j] = A[j][i] for all elements.

2. What is the condition for a matrix to be symmetric?

The matrix must be square (same number of rows and columns) and A[i][j] = A[j][i] for all i and j.

3. Can a non-square matrix be symmetric?

No, a non-square matrix cannot be symmetric because transposing a non-square matrix changes its dimensions, so it cannot equal itself.

4. What is the time complexity of checking symmetry?

The time complexity is O(n²), where n is the size of the matrix. We need to check each element once.

5. What is the difference between symmetric and skew-symmetric?

Symmetric: A[i][j] = A[j][i]
Skew-Symmetric: A[i][j] = -A[j][i] and diagonal elements are zero.

💡 Tip: When checking for symmetry, you only need to check the upper triangle (i < j) to save time. The diagonal and lower triangle are automatically verified.

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