Matrix Multiplication in Java

What is matrix multiplication

Matrix multiplication combines two matrices (2D arrays) into a new one. Each result cell is built by taking a row from the first matrix, a column from the second, multiplying matching elements, and adding them up. It is a classic exercise in nested loops and 2D arrays.

The multiplication rule

For 2×2 matrices A and B, the top-left result cell is: (A's first row) times (B's first column):

c[0][0] = a[0][0]*b[0][0] + a[0][1]*b[1][0]

You repeat this pattern for every result cell, each pairing one row of A with one column of B.

The Java program

public class MatrixMul {
    public static void main(String[] args) {
        int[][] a = {{1, 2}, {3, 4}};
        int[][] b = {{5, 6}, {7, 8}};
        int[][] c = new int[2][2];

        for (int i = 0; i < 2; i++) {          // each row of A
            for (int j = 0; j < 2; j++) {      // each col of B
                for (int k = 0; k < 2; k++) {  // sum the products
                    c[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        for (int[] row : c) {
            for (int x : row) System.out.print(x + " ");
            System.out.println();
        }
    }
}

Three loops: i and j pick the result cell, and k adds up the products of A's row and B's column for that cell.

Dry run

When it is valid

Two matrices can be multiplied only if the columns of the first equal the rows of the second. An m×n matrix times an n×p matrix gives an m×p result. If the inner dimensions do not match, multiplication is not defined.

Common mistakes

• Not initialising the result. Result cells must start at 0 before adding products (Java arrays do this by default).
• Wrong index order. Use a[i][k] * b[k][j] — the shared index is k.
• Mismatched dimensions. Columns of A must equal rows of B.
• Expecting A×B = B×A. Matrix multiplication is not commutative.

Summary

• Each result cell is a row of A dotted with a column of B.
• Use three nested loops: two for the cell, one to sum products.
• Valid only when A's columns equal B's rows.
• An m×n times n×p matrix gives an m×p result.
• Matrix multiplication is not commutative.

Next, practise 2D arrays in Java arrays, or nested loops in Floyd's triangle.