using System;

​

class LinearEquations

{

    // Function to calculate the determinant of a 3x3 matrix

    static double Determinant3x3(double[,] matrix)

    {

        return matrix[0, 0] * (matrix[1, 1] * matrix[2, 2] - matrix[1, 2] * matrix[2, 1])

             - matrix[0, 1] * (matrix[1, 0] * matrix[2, 2] - matrix[1, 2] * matrix[2, 0])

             + matrix[0, 2] * (matrix[1, 0] * matrix[2, 1] - matrix[1, 1] * matrix[2, 0]);

    }

​

    // Replace a column in a 3x3 matrix with a vector

    static double[,] ReplaceColumn(double[,] matrix, double[] column, int columnIndex)

    {

        double[,] tempMatrix = (double[,])matrix.Clone();

        for (int i = 0; i < 3; i++)

        {

            tempMatrix[i, columnIndex] = column[i];

        }

        return tempMatrix;

    }

​

    // Solve system using Cramer's Rule

    static void SolveCramer()

    {

        double[,] A = {

            { 10, 40, 70 },

            { 20, 50, 80 },

            { 30, 60, 80 }

        };

        double[] B = { 300, 360, 390 };

​

        double detA = Determinant3x3(A);

​

        if (detA == 0)

        {

            Console.WriteLine("Matrix is singular. Cramer's method cannot be applied.");

            return;

        }

​

        double[] solutions = new double[3];

        for (int i = 0; i < 3; i++)

        {

            double[,] Ai = ReplaceColumn(A, B, i);

            double detAi = Determinant3x3(Ai);

            solutions[i] = detAi / detA;

        }

​

        Console.WriteLine("Solution using Cramer's Method:");

        Console.WriteLine($"x1 = {solutions[0]}, x2 = {solutions[1]}, x3 = {solutions[2]}");

    }

​

    // Gaussian elimination method to solve a system of equations

    static double[] GaussianElimination(double[,] matrix, double[] vector)

    {

        int n = vector.Length;

​

        for (int i = 0; i < n; i++)

        {

            // Pivot

            for (int k = i + 1; k < n; k++)

            {

                double factor = matrix[k, i] / matrix[i, i];

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

                {

                    matrix[k, j] -= factor * matrix[i, j];

                }

                vector[k] -= factor * vector[i];

            }

        }

​

        // Back substitution

        double[] solution = new double[n];

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

        {

            solution[i] = vector[i];

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

            {

                solution[i] -= matrix[i, j] * solution[j];

            }

            solution[i] /= matrix[i, i];

        }

​

        return solution;

    }

​

    // Solve using Gaussian Elimination for Exercise 1

    static void SolveGaussianEx1()

    {

        double[,] A = {

            { 10, 40, 70 },

            { 20, 50, 80 },

            { 30, 60, 80 }

        };

        double[] B = { 300, 360, 390 };

​

        double[] solution = GaussianElimination(A, B);

​

        Console.WriteLine("Solution using Gaussian Elimination:");

        Console.WriteLine($"x1 = {solution[0]}, x2 = {solution[1]}, x3 = {solution[2]}");

    }

​

    // Solve using Gaussian Elimination for Exercise 3

    static void SolveGaussianEx3()

    {

        double[,] A = {

            { 0, 2, 2 },

            { 3, 3, 0 },

            { 1, 0, 1 }

        };

        double[] B = { 1, 3, 2 };

​

        double detA = Determinant3x3(A);

​

        if (detA == 0)

        {

            Console.WriteLine("Matrix A is singular. No unique solution exists.");

            return;

        }

​

        double[] solution = GaussianElimination(A, B);

​

        Console.WriteLine("Solution for Exercise 3 using Gaussian Elimination:");

        Console.WriteLine($"x1 = {solution[0]}, x2 = {solution[1]}, x3 = {solution[2]}");

    }

​

    static void Main()

    {

        Console.WriteLine("=== Exercise 1 ===");

        SolveCramer();

        SolveGaussianEx1();

​

        Console.WriteLine("\n=== Exercise 3 ===");

        SolveGaussianEx3();

    }

}

​