Example: Inverse of a 3x3 Matrix in C

// Inverse of a 3x3 Matrix
#include <stdio.h>
#include <stdlib.h> // For exit()

// Function to display a 3x3 matrix
void displayMatrix(double mat[3][3]) {
    for (int i = 0; i < 3; i++) {
        for (int j = 0; j < 3; j++) {
            printf("%10.4f ", mat[i][j]);
        }
        printf("\n");
    }
    printf("\n");
}

// Function to calculate the determinant of a 2x2 matrix
double determinant2x2(double a, double b, double c, double d) {
    return a * d - b * c;
}

// Function to calculate the determinant of a 3x3 matrix
double determinant3x3(double mat[3][3]) {
    double det = 0;
    // Formula for 3x3 determinant:
    // a(ei - fh) - b(di - fg) + c(dh - eg)
    det = mat[0][0] * determinant2x2(mat[1][1], mat[1][2], mat[2][1], mat[2][2]) -
          mat[0][1] * determinant2x2(mat[1][0], mat[1][2], mat[2][0], mat[2][2]) +
          mat[0][2] * determinant2x2(mat[1][0], mat[1][1], mat[2][0], mat[2][1]);
    return det;
}

// Function to find the inverse of a 3x3 matrix
void findInverse(double mat[3][3], double inverseMat[3][3]) {
    double det = determinant3x3(mat);

    // Check if determinant is zero (matrix is singular)
    if (det == 0) {
        printf("Determinant is 0, inverse does not exist for this matrix.\n");
        exit(EXIT_FAILURE);
    }

    double adj[3][3]; // Adjoint matrix
    double cofactor[3][3]; // Cofactor matrix (will be transposed to form adjoint)

    // Calculate cofactor matrix
    for (int i = 0; i < 3; i++) {
        for (int j = 0; j < 3; j++) {
            // Create a temporary 2x2 submatrix
            double submatrix[2][2];
            int r = 0, c = 0;
            for (int row = 0; row < 3; row++) {
                for (int col = 0; col < 3; col++) {
                    if (row != i && col != j) {
                        submatrix[r][c++] = mat[row][col];
                        if (c == 2) {
                            c = 0;
                            r++;
                        }
                    }
                }
            }
            // Calculate determinant of 2x2 submatrix and apply sign
            cofactor[i][j] = determinant2x2(submatrix[0][0], submatrix[0][1], submatrix[1][0], submatrix[1][1]);
            if ((i + j) % 2 == 1) { // If (i+j) is odd, multiply by -1
                cofactor[i][j] = -cofactor[i][j];
            }
        }
    }

    // Transpose the cofactor matrix to get the adjoint matrix
    for (int i = 0; i < 3; i++) {
        for (int j = 0; j < 3; j++) {
            adj[j][i] = cofactor[i][j]; // Note: adj[j][i] = cofactor[i][j] for transpose
        }
    }

    // Calculate the inverse matrix (adjugate / determinant)
    for (int i = 0; i < 3; i++) {
        for (int j = 0; j < 3; j++) {
            inverseMat[i][j] = adj[i][j] / det;
        }
    }
}

int main() {
    // Step 1: Initialize the 3x3 matrix
    double matrix[3][3] = {
        {1, 2, 3},
        {0, 1, 4},
        {5, 6, 0}
    };

    double inverse[3][3];

    // Step 2: Display the original matrix
    printf("Original 3x3 Matrix:\n");
    displayMatrix(matrix);

    // Step 3: Find and display the inverse matrix
    findInverse(matrix, inverse);

    printf("Inverse of the 3x3 Matrix:\n");
    displayMatrix(inverse);

    return 0;
}