Finding the matrix row reduction using Gaussian elimination.

Here’s an example Java program that implements Gaussian elimination to find the row-reduced echelon form of a matrix:

import java.util.Arrays;

public class GaussianElimination {

    public static void main(String[] args) {

        double[][] matrix = {

            {2, -3, 1, 4},

            {1, 2, -1, 3},

            {4, -1, 3, 2}

        };     

        int rows = matrix.length;

        int cols = matrix[0].length;      

        System.out.println("Original Matrix:");

        printMatrix(matrix);      

        rowReduction(matrix, rows, cols);

        System.out.println("Row-Reduced Echelon Form:");

        printMatrix(matrix);

    }

    public static void rowReduction(double[][] matrix, int rows, int cols) {

        int lead = 0;      

        for (int r = 0; r < rows; r++) {

            if (lead >= cols)

                break;           

            int i = r;

            while (matrix[i][lead] == 0) {

                i++;

                if (i == rows) {

                    i = r;

                    lead++;

                    if (lead == cols)

                        return;

                }

            }           

            double[] temp = matrix[r];

            matrix[r] = matrix[i];

            matrix[i] = temp;

            double lv = matrix[r][lead];

            for (int j = 0; j < cols; j++)

                matrix[r][j] /= lv;

            for (int k = 0; k < rows; k++) {

                if (k != r) {

                    double factor = matrix[k][lead];

                    for (int j = 0; j < cols; j++)

                        matrix[k][j] -= factor * matrix[r][j];

                }

            }

            lead++;

        }

    } 

    public static void printMatrix(double[][] matrix) {

        for (double[] row : matrix) {

            System.out.println(Arrays.toString(row));

        }

        System.out.println();

    }

}

In this example, the program takes a 2D array representing the matrix and performs Gaussian elimination to obtain the row-reduced echelon form. The `rowReduction` method implements the algorithm, and the `printMatrix` method is used to display the matrix before and after the row reduction.

The program uses a hardcoded matrix for simplicity, but you can modify it to accept user input or generate random matrices if needed.