Example: Matrix Inverse using Determinant and Adjoint in C

// Matrix Inverse using Determinant and Adjoint
#include <stdio.h>

#define N 3 // Define the size of the matrix (N x N)

// Function to get cofactor of matrix[p][q] in temp[][]
// n is current dimension of matrix
void getCofactor(double mat[N][N], double temp[N][N], int p, int q, int n) {
    int i = 0, j = 0;
    // Looping for each element of the matrix
    for (int row = 0; row < n; row++) {
        for (int col = 0; col < n; col++) {
            // Copying into temporary matrix only those elements which are not in given row and column
            if (row != p && col != q) {
                temp[i][j++] = mat[row][col];
                // Row is full, so increment row index and reset col index
                if (j == n - 1) {
                    j = 0;
                    i++;
                }
            }
        }
    }
}

// Recursive function for finding determinant of matrix.
// n is current dimension of mat[][].
double determinant(double mat[N][N], int n) {
    double D = 0; // Initialize result

    // Base case : if matrix contains single element
    if (n == 1)
        return mat[0][0];

    double temp[N][N]; // To store cofactors
    int sign = 1;      // To store sign multiplier

    // Iterate for each element of the first row
    for (int f = 0; f < n; f++) {
        // Getting Cofactor of mat[0][f]
        getCofactor(mat, temp, 0, f, n);
        D += sign * mat[0][f] * determinant(temp, n - 1);

        // Terms are to be added with alternate signs
        sign = -sign;
    }
    return D;
}

// Function to calculate and store adjoint of matrix[N][N] in adj[N][N]
void adjoint(double mat[N][N], double adj[N][N]) {
    if (N == 1) {
        adj[0][0] = 1;
        return;
    }

    int sign = 1;
    double temp[N][N];

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            // Get cofactor of mat[i][j]
            getCofactor(mat, temp, i, j, N);

            // sign is (-1)^(i+j)
            sign = ((i + j) % 2 == 0) ? 1 : -1;

            // Transpose of cofactor matrix is adjoint
            adj[j][i] = (sign) * (determinant(temp, N - 1));
        }
    }
}

// Function to calculate and store inverse, returns false if
// matrix is singular
int inverse(double mat[N][N], double inv[N][N]) {
    // Find determinant of matrix
    double det = determinant(mat, N);
    if (det == 0) {
        printf("Singular matrix, cannot find its inverse.\n");
        return 0;
    }

    // Find adjoint
    double adj[N][N];
    adjoint(mat, adj);

    // Find inverse using formula "inverse(A) = adj(A)/det(A)"
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            inv[i][j] = adj[i][j] / det;
        }
    }
    return 1;
}

// Generic function to display a matrix
void display(double mat[N][N]) {
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            printf("%8.4f ", mat[i][j]);
        }
        printf("\n");
    }
}

int main() {
    // Step 1: Define the matrix for which to find the inverse
    double mat[N][N] = {{1, 2, 3},
                        {4, 5, 6},
                        {7, 8, 9}}; // This matrix is singular (determinant is 0)

    double mat2[N][N] = {{1, 2, 3},
                         {0, 1, 4},
                         {5, 6, 0}}; // This matrix is non-singular

    double inverseMat[N][N];

    printf("Original Matrix 1:\n");
    display(mat);

    // Step 2: Attempt to find the inverse of the first matrix
    if (inverse(mat, inverseMat)) {
        printf("\nInverse of Matrix 1:\n");
        display(inverseMat);
    } else {
        printf("\nMatrix 1 is singular, inverse does not exist.\n");
    }

    printf("\n-------------------------\n");

    printf("\nOriginal Matrix 2:\n");
    display(mat2);

    // Step 3: Attempt to find the inverse of the second matrix
    if (inverse(mat2, inverseMat)) {
        printf("\nInverse of Matrix 2:\n");
        display(inverseMat);
    } else {
        printf("\nMatrix 2 is singular, inverse does not exist.\n");
    }

    return 0;
}