Write A Program For Matrix Multiplication In C

Understanding matrix multiplication is fundamental in various computational fields, from computer graphics to data analysis. In this article, you will learn how to implement matrix multiplication in C, focusing on the core logic and practical considerations.

Problem Statement

Matrix multiplication involves taking two matrices and producing a third matrix. A crucial condition for multiplication is that the number of columns in the first matrix must equal the number of rows in the second matrix. If matrix A is of dimension m x n and matrix B is of dimension n x p, their product C will be a matrix of dimension m x p. Each element C[i][j] is calculated as the sum of the products of corresponding elements from row i of matrix A and column j of matrix B. This operation is vital for transformations, solving linear equations, and many algorithms in scientific computing.

Example

Consider two matrices, A and B:

Matrix A (2x3):

1 2 3
4 5 6

Matrix B (3x2):

7 8
9 1
2 3

The result of A * B will be a 2x2 matrix:

(1*7 + 2*9 + 3*2)   (1*8 + 2*1 + 3*3)
(4*7 + 5*9 + 6*2)   (4*8 + 5*1 + 6*3)

Resultant Matrix C (2x2):

31  19
85  55

Background & Knowledge Prerequisites

To understand and implement matrix multiplication in C, readers should have a basic understanding of:

• C Programming Fundamentals: Variables, data types, and basic input/output operations.
• Arrays: How to declare, initialize, and access elements in one-dimensional and two-dimensional arrays.
• Loops: for loops are extensively used for iterating through matrix elements.
• Functions (Optional but Recommended): While not strictly necessary for a simple example, understanding functions helps in organizing larger programs.

Use Cases or Case Studies

Matrix multiplication is a cornerstone operation across many disciplines:

• Computer Graphics: Used for transforming objects (scaling, rotation, translation) in 2D and 3D space. Each transformation can be represented as a matrix, and applying multiple transformations involves multiplying these matrices.
• Machine Learning and Deep Learning: Fundamental for neural networks, where weights and inputs are often represented as matrices, and matrix multiplication forms the core of layer computations.
• Image Processing: Filters, convolutions, and various image transformations rely on matrix operations.
• Physics and Engineering: Solving systems of linear equations, simulating physical systems, and performing structural analysis often involve matrix algebra.
• Data Analysis and Statistics: Used in algorithms like Principal Component Analysis (PCA) and various regression models to analyze relationships within datasets.

Solution Approaches

The most common and straightforward approach for matrix multiplication involves using nested loops to implement the mathematical definition.

Standard Iterative Matrix Multiplication

This approach directly translates the definition of matrix multiplication into C code using three nested for loops.

One-line summary: Iterates through rows of the first matrix, columns of the second, and then sums the products of elements from the corresponding row and column.

Code example:

// Matrix Multiplication
#include <stdio.h>

#define ROW1 2
#define COL1 3
#define ROW2 3
#define COL2 2

int main() {
    // Step 1: Declare and initialize matrices
    int matrix1[ROW1][COL1] = {
        {1, 2, 3},
        {4, 5, 6}
    };

    int matrix2[ROW2][COL2] = {
        {7, 8},
        {9, 1},
        {2, 3}
    };

    int resultMatrix[ROW1][COL2]; // Resultant matrix dimensions

    // Step 2: Check for multiplication compatibility
    // Number of columns in matrix1 must be equal to number of rows in matrix2
    if (COL1 != ROW2) {
        printf("Error: Matrices cannot be multiplied. Number of columns in first matrix must equal number of rows in second matrix.\\n");
        return 1; // Indicate error
    }

    // Step 3: Perform matrix multiplication
    printf("Multiplying Matrix 1 (%dx%d) by Matrix 2 (%dx%d):\\n", ROW1, COL1, ROW2, COL2);

    // Loop through rows of result matrix
    for (int i = 0; i < ROW1; i++) {
        // Loop through columns of result matrix
        for (int j = 0; j < COL2; j++) {
            resultMatrix[i][j] = 0; // Initialize element to 0
            // Loop through elements for dot product
            for (int k = 0; k < COL1; k++) { // Or k < ROW2, since COL1 == ROW2
                resultMatrix[i][j] += matrix1[i][k] * matrix2[k][j];
            }
        }
    }

    // Step 4: Display the result matrix
    printf("\\nResultant Matrix (%dx%d):\\n", ROW1, COL2);
    for (int i = 0; i < ROW1; i++) {
        for (int j = 0; j < COL2; j++) {
            printf("%d\\t", resultMatrix[i][j]);
        }
        printf("\\n");
    }

    return 0;
}

Run

Sample output:

Multiplying Matrix 1 (2x3) by Matrix 2 (3x2):

Resultant Matrix (2x2):
31	19	
85	55

Stepwise explanation for clarity:

1. Define Dimensions: #define statements are used to clearly specify the dimensions of the input matrices. This makes the code more readable and easier to modify.
2. Declare Matrices:
   • matrix1 is declared with ROW1 rows and COL1 columns.

1. Compatibility Check: Before multiplication, an if statement verifies if COL1 == ROW2. This is a mathematical requirement for matrix multiplication. If not met, an error message is printed.
2. Outer Loops (i and j):
   • The outermost loop (for (int i = 0; i < ROW1; i++)) iterates through each row of the resultMatrix.

1. Inner Loop (k):
   • The innermost loop (for (int k = 0; k < COL1; k++)) performs the dot product calculation. It iterates COL1 (which is equal to ROW2) times.

1. Display Result: After all calculations are complete, another set of nested loops prints the elements of resultMatrix in a clear format.

Conclusion

Implementing matrix multiplication in C using nested loops is a fundamental exercise that demonstrates array manipulation and iterative logic. While this basic approach is straightforward, understanding its core mechanics is crucial before exploring more advanced and optimized algorithms for larger matrices, such as Strassen's algorithm or parallel processing techniques.

Summary

• Matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second.
• The resultant matrix has dimensions equal to the rows of the first and columns of the second matrix.
• The standard iterative solution uses three nested for loops to calculate each element of the product matrix.
• This operation is foundational in computer graphics, machine learning, and various scientific computing applications.