C Program To Check If A Given Matrix Is An Identity Matrix

An identity matrix is a special square matrix where all the elements on the main diagonal are 1s, and all other elements are 0s. Recognizing an identity matrix is fundamental in linear algebra for various operations. In this article, you will learn how to write a C program to efficiently check if a given matrix is an identity matrix.

Problem Statement

The challenge is to determine whether a user-provided matrix, represented as a 2D array, adheres to the definition of an identity matrix. This involves verifying two key conditions: first, the matrix must be square (number of rows equals number of columns); second, all diagonal elements must be 1, and all non-diagonal elements must be 0.

Example

Consider these two matrices:

Matrix A (Identity Matrix):

1 0 0
0 1 0
0 0 1

Expected output: "The matrix is an Identity Matrix."

Matrix B (Non-Identity Matrix):

1 0 0
0 2 0
0 0 1

Expected output: "The matrix is NOT an Identity Matrix." (because the element at [1][1] is 2, not 1)

Background & Knowledge Prerequisites

To understand and implement the solution, a basic understanding of the following C programming concepts is beneficial:

• Arrays: Specifically, two-dimensional arrays for representing matrices.
• Loops: for loops for iterating through matrix elements.
• Conditional Statements: if-else statements for checking conditions.
• Functions: For better code organization (though a single main function solution is also acceptable for simplicity).
• Basic Matrix Concepts: Knowledge of rows, columns, and diagonal elements.

Use Cases or Case Studies

Checking for an identity matrix is crucial in several computational and mathematical contexts:

• Linear Algebra Operations: Inverting matrices, solving systems of linear equations, and performing matrix transformations often involve identity matrices.
• Computer Graphics: Identity matrices are used as a base for transformations (translation, rotation, scaling) in 2D and 3D graphics, representing no change to an object.
• Testing Matrix Libraries: When developing or testing functions for matrix manipulation, verifying the output against an identity matrix is a common unit test.
• Cryptographic Algorithms: Some cryptographic methods use matrix operations where the identity matrix plays a specific role in certain transformations or key generation.
• Network Analysis: In some graph theory applications, adjacency matrices might be compared against identity-like structures to understand node relationships.

Solution Approaches

The most straightforward and efficient approach to check for an identity matrix involves iterating through all its elements and applying the definition directly.

Iterative Check for Identity Matrix

This approach systematically checks two conditions: first, that the matrix is square, and second, that every element conforms to the identity matrix definition.

One-line summary

Iterate through the matrix, checking if diagonal elements are 1 and non-diagonal elements are 0, after confirming the matrix is square.

Code example

// Check if a matrix is an Identity Matrix
#include <stdio.h>
#include <stdbool.h> // For using boolean type

// Function to check if a matrix is an identity matrix
bool isIdentityMatrix(int matrix[10][10], int rows, int cols) {
    // Step 1: Check if the matrix is square
    if (rows != cols) {
        return false; // Not a square matrix, so cannot be an identity matrix
    }

    // Step 2: Iterate through all elements of the matrix
    for (int i = 0; i < rows; i++) {
        for (int j = 0; j < cols; j++) {
            // Step 3: Check diagonal elements
            if (i == j) { // Current element is on the main diagonal
                if (matrix[i][j] != 1) {
                    return false; // Diagonal element is not 1
                }
            } else { // Current element is not on the main diagonal
                // Step 4: Check non-diagonal elements
                if (matrix[i][j] != 0) {
                    return false; // Non-diagonal element is not 0
                }
            }
        }
    }

    // Step 5: If all checks pass, it's an identity matrix
    return true;
}

int main() {
    int rows1 = 3, cols1 = 3;
    int matrix1[10][10] = {
        {1, 0, 0},
        {0, 1, 0},
        {0, 0, 1}
    };

    printf("Matrix 1:\\n");
    for (int i = 0; i < rows1; i++) {
        for (int j = 0; j < cols1; j++) {
            printf("%d ", matrix1[i][j]);
        }
        printf("\\n");
    }

    if (isIdentityMatrix(matrix1, rows1, cols1)) {
        printf("The matrix 1 is an Identity Matrix.\\n\\n");
    } else {
        printf("The matrix 1 is NOT an Identity Matrix.\\n\\n");
    }

    int rows2 = 3, cols2 = 3;
    int matrix2[10][10] = {
        {1, 0, 0},
        {0, 2, 0},
        {0, 0, 1}
    };

    printf("Matrix 2:\\n");
    for (int i = 0; i < rows2; i++) {
        for (int j = 0; j < cols2; j++) {
            printf("%d ", matrix2[i][j]);
        }
        printf("\\n");
    }

    if (isIdentityMatrix(matrix2, rows2, cols2)) {
        printf("The matrix 2 is an Identity Matrix.\\n\\n");
    } else {
        printf("The matrix 2 is NOT an Identity Matrix.\\n\\n");
    }

    int rows3 = 2, cols3 = 3; // Non-square matrix
    int matrix3[10][10] = {
        {1, 0, 0},
        {0, 1, 0}
    };

    printf("Matrix 3:\\n");
    for (int i = 0; i < rows3; i++) {
        for (int j = 0; j < cols3; j++) {
            printf("%d ", matrix3[i][j]);
        }
        printf("\\n");
    }

    if (isIdentityMatrix(matrix3, rows3, cols3)) {
        printf("The matrix 3 is an Identity Matrix.\\n\\n");
    } else {
        printf("The matrix 3 is NOT an Identity Matrix.\\n\\n");
    }

    return 0;
}

Run

Sample output

Matrix 1:
1 0 0 
0 1 0 
0 0 1 
The matrix 1 is an Identity Matrix.

Matrix 2:
1 0 0 
0 2 0 
0 0 1 
The matrix 2 is NOT an Identity Matrix.

Matrix 3:
1 0 0 
0 1 0 
The matrix 3 is NOT an Identity Matrix.

Stepwise explanation

1. Include Headers: stdio.h for input/output and stdbool.h for using true/false.
2. isIdentityMatrix Function:
   • Takes the matrix, number of rows, and number of columns as input.

Square Matrix Check: The first and most critical check is if (rows != cols). An identity matrix must always be square. If it's not, the function immediately returns false.

Element Iteration: Uses nested for loops to traverse every element matrix[i][j] of the matrix.

Diagonal Element Check: Inside the loops, if (i == j) identifies elements on the main diagonal. For these elements, it checks if (matrix[i][j] != 1). If any diagonal element is not 1, the matrix is not an identity matrix, and false is returned.

Non-Diagonal Element Check: For elements where i != j (non-diagonal elements), it checks if (matrix[i][j] != 0). If any non-diagonal element is not 0, it's not an identity matrix, and false is returned.

Return True: If the loops complete without any of the above conditions returning false, it means all elements satisfy the identity matrix criteria, so the function returns true.

1. main Function:
   • Demonstrates the isIdentityMatrix function with three different matrices: a valid identity matrix, a non-identity square matrix, and a non-square matrix.

Prints the matrix and the result of the check for each example.

Conclusion

Understanding and implementing a check for an identity matrix is a fundamental skill in C programming, particularly when dealing with mathematical computations. The iterative approach provides a clear and efficient way to verify if a given matrix adheres to the specific properties of an identity matrix by checking its dimensions and every individual element.

Summary

• An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere.
• To check, first ensure the matrix is square (rows == columns).
• Then, iterate through the matrix:
• Diagonal elements (i == j) must be 1.
• Non-diagonal elements (i != j) must be 0.
• Any deviation from these rules means it's not an identity matrix.