# C++ - Solving Linear Equation (3 X 3) Using Matrix

Assume that you have the following 3 equations and you have to find the value of X, Y and Z using Matrices.

Problem:

X -   Y + 2Z = 2

2X -  3Y -  Z = 5

3X + .5Y +  Z = 3

X =  1.16

Y = -0.89

Z = -0.02

This problem has in the matrix form - 3 X 3 square matrix.

Lets form an equation:

ax = b

where a

|       1.00      -1.00       2.00 |

|       2.00      -3.00      -1.00 |

|       3.00       0.50       1.00 |

and b

|       2.00 |

|       5.00 |

|       3.00 |

If we need to find out the X in equation ax = b, then apply the formula:

x = InverseMatrix of A multiplied by B.

Given the matrix A, we have to find out its inverse form. Then do the multiplication with B to get the answer B.

The following program does the work for you. Look at the program and output.

## Source Code

#if !defined(MATRIX_H)

#define MATRIX_H

#include <stdio.h>

#include <iostream>

#include <tchar.h>

#include <math.h>

class CMatrix

{

private:

int m_rows;

int m_cols;

char m_name[128];

CMatrix();

public:

double **m_pData;

CMatrix(const char *name, int rows, int cols) : m_rows(rows), m_cols(cols)

{

strcpy(m_name, name);

m_pData = new double*[m_rows];

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

m_pData[i] = new double[m_cols];

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

{

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

{

m_pData[i][j] = 0.0;

}

}

}

CMatrix(const CMatrix &other)

{

strcpy(m_name, other.m_name);

m_rows = other.m_rows;

m_cols = other.m_cols;

m_pData = new double*[m_rows];

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

m_pData[i] = new double[m_cols];

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

{

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

{

m_pData[i][j] = other.m_pData[i][j];

}

}

}

~CMatrix()

{

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

delete [] m_pData[i];

delete [] m_pData;

m_rows = m_cols = 0;

}

void SetName(const char *name) { strcpy(m_name, name); }

const char* GetName() const { return m_name; }

void GetInput()

{

std::cin >> *this;

}

void FillSimulatedInput()

{

static int factor1 = 1, factor2 = 2;

std::cout << "\n\nEnter Input For Matrix : " << m_name << " Rows: " << m_rows << " Cols: " << m_cols << "\n";

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

{

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

{

std::cout << "Input For Row: " << i + 1 << " Col: " << j + 1 << " = ";

int data = ((i + 1) * factor1) + (j + 1) * factor2;

m_pData[i][j] = data / 10.2;

std::cout << m_pData[i][j] << "\n";

factor1 += (rand() % 4);

factor2 += (rand() % 3);

}

std::cout << "\n";

}

std::cout << "\n";

}

double Determinant()

{

double det = 0;

double **pd = m_pData;

switch(m_rows)

{

case 2:

{

det = pd[0][0] * pd[1][1] - pd[0][1] * pd[1][0];

return det;

}

break;

case 3:

{

/***

a b c

d e f

g h i

a b c a b c

d e f d e f

g h i g h i

// det (A) = aei + bfg + cdh - afh - bdi - ceg.

***/

double a = pd[0][0];

double b = pd[0][1];

double c = pd[0][2];

double d = pd[1][0];

double e = pd[1][1];

double f = pd[1][2];

double g = pd[2][0];

double h = pd[2][1];

double i = pd[2][2];

double det = (a*e*i + b*f*g + c*d*h); // - a*f*h - b*d*i - c*e*g);

det = det - a*f*h;

det = det - b*d*i;

det = det - c*e*g;

//std::cout << *this;

//std::cout << "deter: " << det << " \n";

return det;

}

break;

case 4:

{

CMatrix *temp[4];

for(int i = 0; i < 4; i++)

temp[i] = new CMatrix("", 3,3);

for(int k = 0; k < 4; k++)

{

for(int i = 1; i < 4; i++)

{

int j1 = 0;

for(int j = 0; j < 4; j++)

{

if(k == j)

continue;

temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];

}

}

}

double det = this->m_pData[0][0] * temp[0]->Determinant() -

this->m_pData[0][1] * temp[1]->Determinant() +

this->m_pData[0][2] * temp[2]->Determinant() -

this->m_pData[0][3] * temp[3]->Determinant();

return det;

}

break;

case 5:

{

CMatrix *temp[5];

for(int i = 0; i < 5; i++)

temp[i] = new CMatrix("", 4,4);

for(int k = 0; k < 5; k++)

{

for(int i = 1; i < 5; i++)

{

int j1 = 0;

for(int j = 0; j < 5; j++)

{

if(k == j)

continue;

temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];

}

}

}

double det = this->m_pData[0][0] * temp[0]->Determinant() -

this->m_pData[0][1] * temp[1]->Determinant() +

this->m_pData[0][2] * temp[2]->Determinant() -

this->m_pData[0][3] * temp[3]->Determinant() +

this->m_pData[0][4] * temp[4]->Determinant();

return det;

}

case 6:

case 7:

case 8:

case 9:

case 10:

case 11:

case 12:

default:

{

int DIM = m_rows;

CMatrix **temp = new CMatrix*[DIM];

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

temp[i] = new CMatrix("", DIM - 1,DIM - 1);

for(int k = 0; k < DIM; k++)

{

for(int i = 1; i < DIM; i++)

{

int j1 = 0;

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

{

if(k == j)

continue;

temp[k]->m_pData[i-1][j1++] = this->m_pData[i][j];

}

}

}

double det = 0;

for(int k = 0; k < DIM; k++)

{

if( (k %2) == 0)

det = det + (this->m_pData[0][k] * temp[k]->Determinant());

else

det = det - (this->m_pData[0][k] * temp[k]->Determinant());

}

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

delete temp[i];

delete [] temp;

return det;

}

break;

}

}

CMatrix& operator = (const CMatrix &other)

{

if( this->m_rows != other.m_rows ||

this->m_cols != other.m_cols)

{

std::cout << "WARNING: Assignment is taking place with by changing the number of rows and columns of the matrix";

}

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

delete [] m_pData[i];

delete [] m_pData;

m_rows = m_cols = 0;

strcpy(m_name, other.m_name);

m_rows = other.m_rows;

m_cols = other.m_cols;

m_pData = new double*[m_rows];

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

m_pData[i] = new double[m_cols];

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

{

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

{

m_pData[i][j] = other.m_pData[i][j];

}

}

return *this;

}

CMatrix CoFactor()

{

CMatrix cofactor("COF", m_rows, m_cols);

if(m_rows != m_cols)

return cofactor;

if(m_rows < 2)

return cofactor;

else if(m_rows == 2)

{

cofactor.m_pData[0][0] = m_pData[1][1];

cofactor.m_pData[0][1] = -m_pData[1][0];

cofactor.m_pData[1][0] = -m_pData[0][1];

cofactor.m_pData[1][1] = m_pData[0][0];

return cofactor;

}

else if(m_rows >= 3)

{

int DIM = m_rows;

CMatrix ***temp = new CMatrix**[DIM];

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

temp[i] = new CMatrix*[DIM];

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

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

temp[i][j] = new CMatrix("", DIM - 1,DIM - 1);

for(int k1 = 0; k1 < DIM; k1++)

{

for(int k2 = 0; k2 < DIM; k2++)

{

int i1 = 0;

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

{

int j1 = 0;

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

{

if(k1 == i || k2 == j)

continue;

temp[k1][k2]->m_pData[i1][j1++] = this->m_pData[i][j];

}

if(k1 != i)

i1++;

}

}

}

bool flagPositive = true;

for(int k1 = 0; k1 < DIM; k1++)

{

flagPositive = ( (k1 % 2) == 0);

for(int k2 = 0; k2 < DIM; k2++)

{

if(flagPositive == true)

{

cofactor.m_pData[k1][k2] = temp[k1][k2]->Determinant();

flagPositive = false;

}

else

{

cofactor.m_pData[k1][k2] = -temp[k1][k2]->Determinant();

flagPositive = true;

}

}

}

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

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

delete temp[i][j];

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

delete [] temp[i];

delete [] temp;

}

return cofactor;

}

{

CMatrix cofactor("COF", m_rows, m_cols);

if(m_rows != m_cols)

cofactor = this->CoFactor();

// adjoint is transpose of a cofactor of a matrix

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

{

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

{

}

}

}

CMatrix Transpose()

{

CMatrix trans("TR", m_cols, m_rows);

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

{

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

{

trans.m_pData[j][i] = m_pData[i][j];

}

}

return trans;

}

CMatrix Inverse()

{

CMatrix cofactor("COF", m_rows, m_cols);

CMatrix inv("INV", m_rows, m_cols);

if(m_rows != m_cols)

return inv;

// to find out Determinant

double det = Determinant();

cofactor = this->CoFactor();

// inv = transpose of cofactor / Determinant

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

{

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

{

inv.m_pData[j][i] = cofactor.m_pData[i][j] / det;

}

}

return inv;

}

CMatrix operator + (const CMatrix &other)

{

if( this->m_rows != other.m_rows ||

this->m_cols != other.m_cols)

{

std::cout << "Addition could not take place because number of rows and columns are different between the two matrices";

return *this;

}

CMatrix result("", m_rows, m_cols);

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

{

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

{

result.m_pData[i][j] = this->m_pData[i][j] + other.m_pData[i][j];

}

}

return result;

}

CMatrix operator - (const CMatrix &other)

{

if( this->m_rows != other.m_rows ||

this->m_cols != other.m_cols)

{

std::cout << "Subtraction could not take place because number of rows and columns are different between the two matrices";

return *this;

}

CMatrix result("", m_rows, m_cols);

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

{

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

{

result.m_pData[i][j] = this->m_pData[i][j] - other.m_pData[i][j];

}

}

return result;

}

CMatrix operator * (const CMatrix &other)

{

if( this->m_cols != other.m_rows)

{

std::cout << "Multiplication could not take place because number of columns of 1st Matrix and number of rows in 2nd Matrix are different";

return *this;

}

CMatrix result("", this->m_rows, other.m_cols);

for(int i = 0; i < this->m_rows; i++)

{

for(int j = 0; j < other.m_cols; j++)

{

for(int k = 0; k < this->m_cols; k++)

{

result.m_pData[i][j] += this->m_pData[i][k] * other.m_pData[k][j];

}

}

}

return result;

}

bool operator == (const CMatrix &other)

{

if( this->m_rows != other.m_rows ||

this->m_cols != other.m_cols)

{

std::cout << "Comparision could not take place because number of rows and columns are different between the two matrices";

return false;

}

CMatrix result("", m_rows, m_cols);

bool bEqual = true;

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

{

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

{

if(this->m_pData[i][j] != other.m_pData[i][j])

bEqual = false;

}

}

return bEqual;

}

friend std::istream& operator >> (std::istream &is, CMatrix &m);

friend std::ostream& operator << (std::ostream &os, const CMatrix &m);

};

std::istream& operator >> (std::istream &is, CMatrix &m)

{

std::cout << "\n\nEnter Input For Matrix : " << m.m_name << " Rows: " << m.m_rows << " Cols: " << m.m_cols << "\n";

for(int i = 0; i < m.m_rows; i++)

{

for(int j = 0; j < m.m_cols; j++)

{

std::cout << "Input For Row: " << i + 1 << " Col: " << j + 1 << " = ";

is >> m.m_pData[i][j];

}

std::cout << "\n";

}

std::cout << "\n";

return is;

}

std::ostream& operator << (std::ostream &os,const CMatrix &m)

{

os << "\n\nMatrix : " << m.m_name << " Rows: " << m.m_rows << " Cols: " << m.m_cols << "\n\n";

for(int i = 0; i < m.m_rows; i++)

{

os << " | ";

for(int j = 0; j < m.m_cols; j++)

{

char buf[32];

double data = m.m_pData[i][j];

if( m.m_pData[i][j] > -0.00001 &&

m.m_pData[i][j] < 0.00001)

data = 0;

sprintf(buf, "%10.2lf ", data);

os <<  buf;

}

os << "|\n";

}

os << "\n\n";

return os;

}

#endif

int main()

{

CMatrix a("A", 6,6);

CMatrix b("B", 6,1);

a.FillSimulatedInput();

b.FillSimulatedInput();

std::cout << a  << "\n Determinant : ";

std::cout << a.Determinant() << "\n";

std::cout << b  << "\n Determinant : ";

std::cout << b.Determinant() << "\n";

CMatrix ainv = a.Inverse();

CMatrix q = a * ainv;

q.SetName("A * A'");

std::cout << q << "\n";

CMatrix x = ainv * b;

x.SetName("X");

std::cout << x << "\n";

CMatrix y = a * x; // we will get B

y.SetName("Y");

std::cout << y << "\n";

return 0;

}

## Output

Enter Input For Matrix : A Rows: 3 Cols: 3

Input For Row: 1 Col: 1 = 1

Input For Row: 1 Col: 2 = -1

Input For Row: 1 Col: 3 = 2

Input For Row: 2 Col: 1 = 2

Input For Row: 2 Col: 2 = -3

Input For Row: 2 Col: 3 = -1

Input For Row: 3 Col: 1 = 3

Input For Row: 3 Col: 2 = 0.5

Input For Row: 3 Col: 3 = 1

Enter Input For Matrix : B Rows: 3 Cols: 1

Input For Row: 1 Col: 1 = 2

Input For Row: 2 Col: 1 = 5

Input For Row: 3 Col: 1 = 3

Matrix : A Rows: 3 Cols: 3

|       1.00      -1.00       2.00 |

|       2.00      -3.00      -1.00 |

|       3.00       0.50       1.00 |

Determinant : 22.5

Matrix : B Rows: 3 Cols: 1

|       2.00 |

|       5.00 |

|       3.00 |

Determinant : 7.29768e-164

Matrix : A * A' Rows: 3 Cols: 3

|       1.00       0.00       0.00 |

|       0.00       1.00       0.00 |

|       0.00       0.00       1.00 |

Matrix : X Rows: 3 Cols: 1

|       1.16 |

|      -0.89 |

|      -0.02 |

Matrix : Y Rows: 3 Cols: 1

|       2.00 |

|       5.00 |

|       3.00 |

Press any key to continue . . .