1) Determinant 2) Inverse Matrix 3) Gaussian Elimination 4) Iterative technique and calculate using three method to find x , y, z method: cramer's rule, gaussian elimination and inverse matrix.

#include <iostream>

#include <vector>

# GaussianElimination

std::vector<std::vector<float>> GaussianElimination(std::vector<std::vector<float>> matrix)

{

  int rows = matrix.size();

  int cols = matrix[0].size();

  for (int i = 0; i < rows; i++)

  {

    float coefficient1 = matrix[i][i];

    for (int j = 0; j < cols; j++)

    {

      matrix[i][j] /= coefficient1;

    }

    for (int j = 0; j < rows; j++)

    {

      if (j == i) continue;

      float coefficient2 = matrix[j][i];

      for (int k = 0; k < cols; k++)

      {

        matrix[j][k] -=  coefficient2 * matrix[i][k];

      }

    }

  }

  return matrix;

}

void PrintMatrix(std::vector<std::vector<float>> matrix)

{

  for (int i = 0; i < matrix.size(); i++)

  {

    for (int j = 0; j < matrix[0].size(); j++)

    {

      std::cout.width(5); std::cout << matrix[i][j] << "    ";

    }

    std::cout << std::endl;

  }

}

void PrintSE(std::vector<std::vector<float>> matrix)

{

  std::cout << "X = ";

  for (int i = 0; i < matrix.size()-1; i++)

  {

    std::cout << matrix[i][matrix[0].size()-1] << ", ";

  }

  std::cout << matrix[matrix.size() - 1][matrix[0].size() - 1];

  std::cout << std::endl;

}

# Inverse matrix

void PrintInverse(std::vector<std::vector<float>> matrix)

{

  std::cout << "The inverse matrix is\n";

  int rows = matrix.size();

  int cols = matrix[0].size();

  for (int i = 0; i < rows; i++)

  {

    for (int j = rows; j < cols; j++)

    {

      std::cout.width(5); std::cout << matrix[i][j] << "    ";

    }

    std::cout << std::endl;

  }

  std::cout << std::endl;

}

int main()

{

  std::cout << "Result of simultaneous equations with Gaussian Elimination" << std::endl;

  std::vector<std::vector<float>>(matrixA) = { {1, -4, -2, 21}, {2, 1, 2, 3 }, {3, 2, -1, -2} };

  std::vector<std::vector<float>>(ResultMatrixA) = GaussianElimination(matrixA);

  PrintMatrix(ResultMatrixA);

  PrintSE(ResultMatrixA);

  std::cout << std::endl;

  std::cout << std::endl;

  std::cout << "Result of inverse matrix with Gaussian Elimination" << std::endl;

  std::vector<std::vector<float>>(matrixB) = { {-1, 8, -2, 1, 0, 0}, {-6, 49, -10, 0, 1, 0}, {-4, 34, -5, 0, 0, 1} };

  std::vector<std::vector<float>>(ResultMatrixB) = GaussianElimination(matrixB);

  PrintMatrix(ResultMatrixB);

  PrintInverse(ResultMatrixB);

  std::cin.get();

}

# demterminant matrix

from copy import deepcopy

def smaller_matrix(original_matrix, row, column):

    #the new metrix should not affect the original matrix

    new_matrix = deepcopy(original_matrix)

    #the indices to do the removal depend on the original metrix

    new_matrix.remove(original_matrix[row])

    for i in range(len(new_matrix)):

        new_matrix[i].remove(new_matrix[i][column])

    return new_matrix

def determinant(matrix):

    num_rows = len(matrix)

    for row in matrix:

        if len(row) != num_rows:

            print("Not a square matrix.")

            return None

        if len(matrix) == 2:

            simple_determinant = matrix[0][0] *\

                                 matrix[1][1] -\

                                 matrix[1][0] *\

                                 matrix[0][1]

            return simple_determinant

        else:

            # cofactor expansion

            answer = 0

            num_columns = num_rows

            for j in range(num_columns):

                cofactor = (-1) ** (0+j) * matrix[0][j] \

                            * determinant(smaller_matrix(matrix,0,j))

                answer += cofactor

            return answer

a_matrix = [[2, 1, 7, 9], [7, 1, 9, 7],[8, 6, 1, 4], [0, 1, 4, 2]]

print("The answer of matirx is :",determinant(a_matrix))