Matrix multiplication in C involves multiplying two matrices to produce a third matrix. This operation requires the number of columns in the first matrix to equal the number of rows in the second matrix. Here’s a detailed explanation along with a sample implementation.

Explanation of Matrix Multiplication

1. Definition:

   • Matrix multiplication is an operation where two matrices are multiplied to yield a new matrix. The element at the position $C[i][j]$ in the resulting matrix $C$ is calculated by taking the dot product of the $i$-th row of the first matrix $A$ and the $j$-th column of the second matrix $B$.

2. Mathematical Representation:

   • For two matrices $A$ (of size $m \times n$) and $B$ (of size $n \times p$), the resulting matrix $C$ will have the size $m \times p$:
   $$C[i][j] = \sum_{k=0}^{n-1} A[i][k] \cdot B[k][j]$$
   • Here, $i$ is the row index for $A$, $j$ is the column index for $B$, and $k$ is used to iterate over the columns of $A$ and the rows of $B$.

3. Conditions:

   • The number of columns in matrix $A$ must be equal to the number of rows in matrix $B$. If $A$ has dimensions $m \times n$, $B$ must have dimensions $n \times p$.

Sample C Program for Matrix Multiplication

Here is a simple C program that demonstrates how to perform matrix multiplication:

#include <stdio.h>

#define MAX 10 // Maximum size of the matrix

void inputMatrix(int matrix[MAX][MAX], int rows, int cols) {
    printf("Enter elements of the matrix:\n");
    for (int i = 0; i < rows; i++) {
        for (int j = 0; j < cols; j++) {
            printf("Element [%d][%d]: ", i, j);
            scanf("%d", &matrix[i][j]);
        }
    }
}

void printMatrix(int matrix[MAX][MAX], int rows, int cols) {
    printf("The matrix is:\n");
    for (int i = 0; i < rows; i++) {
        for (int j = 0; j < cols; j++) {
            printf("%d ", matrix[i][j]);
        }
        printf("\n");
    }
}

void multiplyMatrices(int A[MAX][MAX], int B[MAX][MAX], int C[MAX][MAX], int rowA, int colA, int colB) {
    for (int i = 0; i < rowA; i++) {
        for (int j = 0; j < colB; j++) {
            C[i][j] = 0; // Initialize the element
            for (int k = 0; k < colA; k++) {
                C[i][j] += A[i][k] * B[k][j]; // Multiply and accumulate
            }
        }
    }
}

int main() {
    int A[MAX][MAX], B[MAX][MAX], C[MAX][MAX];
    int rowA, colA, rowB, colB;

    // Input dimensions for the first matrix
    printf("Enter number of rows and columns for Matrix A: ");
    scanf("%d %d", &rowA, &colA);

    // Input dimensions for the second matrix
    printf("Enter number of rows and columns for Matrix B: ");
    scanf("%d %d", &rowB, &colB);

    // Check if multiplication is possible
    if (colA != rowB) {
        printf("Error: Number of columns in Matrix A must equal number of rows in Matrix B.\n");
        return 1; // Exit with error
    }

    // Input matrices A and B
    printf("Matrix A:\n");
    inputMatrix(A, rowA, colA);
    printf("Matrix B:\n");
    inputMatrix(B, rowB, colB);

    // Multiply the two matrices
    multiplyMatrices(A, B, C, rowA, colA, colB);

    // Print the result
    printf("Resultant Matrix C (A * B):\n");
    printMatrix(C, rowA, colB);

    return 0;
}

Explanation of the Code

1. Header Files:

   • #include <stdio.h>: Includes the standard input-output library.

2. Macro Definition:

   • #define MAX 10: Defines a constant for the maximum size of the matrices.

3. Functions:

   • inputMatrix: This function prompts the user to enter the elements of the matrix.
   • printMatrix: This function prints the matrix to the console.
   • multiplyMatrices: This function takes two matrices (A and B) and computes their product, storing the result in matrix C.

4. main Function:

   • Prompts the user for the number of rows and columns for both matrices.
   • Checks if the multiplication is valid (i.e., the number of columns in A must equal the number of rows in B).
   • Calls inputMatrix to fill matrices A and B.
   • Calls multiplyMatrices to perform the multiplication.
   • Calls printMatrix to display the resultant matrix C.

How to Run the Program

1. Compile the Code: Use a C compiler like gcc to compile the code.

   gcc matrix_multiplication.c -o matrix_multiplication
   

2. Execute the Program:

   ./matrix_multiplication
   

3. Input Data: Follow the prompts to input the dimensions and elements of the matrices.

Example Input/Output

Input:

Enter number of rows and columns for Matrix A: 2 3
Enter number of rows and columns for Matrix B: 3 2
Matrix A:
Element [0][0]: 1
Element [0][1]: 2
Element [0][2]: 3
Element [1][0]: 4
Element [1][1]: 5
Element [1][2]: 6
Matrix B:
Element [0][0]: 7
Element [0][1]: 8
Element [1][0]: 9
Element [1][1]: 10
Element [2][0]: 11
Element [2][1]: 12

Output:

Resultant Matrix C (A * B):
58 64 
139 154 

Conclusion

Matrix multiplication is a fundamental operation in linear algebra and is widely used in various fields, including computer graphics, machine learning, and scientific computing. The above example provides a clear understanding of how to implement matrix multiplication in C. You can extend this program further by adding error handling, supporting larger matrices, or using dynamic memory allocation.