Solving a system of linear equations using matrices.

Here’s an example program in Java that solves a system of linear equations using matrices:

   import java.util.Arrays;

    public class LinearEquationSolver {

    public static void main(String[] args) {

        double[][] coefficients = {{2, 1, -1}, {-3, -1, 2}, {-2, 1, 2}};

        double[] constants = {8, -11, -3};

        double[] solution = solveLinearEquation(coefficients, constants);

        System.out.println("Solution: " + Arrays.toString(solution));

    }

    public static double[] solveLinearEquation(double[][] coefficients, double[] constants) {

        int n = coefficients.length;

        // Augmented matrix

        double[][] augmentedMatrix = new double[n][n + 1];

        // Populate augmented matrix

        for (int i = 0; i < n; i++) {

            for (int j = 0; j < n; j++) {

                augmentedMatrix[i][j] = coefficients[i][j];

            }

            augmentedMatrix[i][n] = constants[i];

        }

        // Perform Gaussian elimination

        for (int i = 0; i < n - 1; i++) {

            if (augmentedMatrix[i][i] == 0) {

                // Swap rows to avoid division by zero

                for (int j = i + 1; j < n; j++) {

                    if (augmentedMatrix[j][i] != 0) {

                        swapRows(augmentedMatrix, i, j);

                        break;

                    }

                }

            }

            for (int j = i + 1; j < n; j++) {

                double ratio = augmentedMatrix[j][i] / augmentedMatrix[i][i];

                for (int k = i; k <= n; k++) {

                    augmentedMatrix[j][k] -= ratio * augmentedMatrix[i][k];

                }

            }

        }

        // Back substitution

        double[] solution = new double[n];

        for (int i = n - 1; i >= 0; i--) {

            double sum = 0;

            for (int j = i + 1; j < n; j++) {

                sum += augmentedMatrix[i][j] * solution[j];

            }

            solution[i] = (augmentedMatrix[i][n] - sum) / augmentedMatrix[i][i];

        }

        return solution;

    }

    public static void swapRows(double[][] matrix, int i, int j) {

        double[] temp = matrix[i];

        matrix[i] = matrix[j];

        matrix[j] = temp;

    }

}

This program solves a system of linear equations represented by the `coefficients` matrix and `constants` array. The `solveLinearEquation` method uses Gaussian elimination and back substitution to solve the system. The solution is then returned as an array of doubles.

In the example given, the program solves the following system of linear equations:

2x + y – z = 8

-3x – y + 2z = -11

-2x + y + 2z = -3

The solution is printed to the console.