Find determinant of a matrix in C++

By Pritam Hande

In this tutorial, we will learn how to find the determinant of a matrix in C++.

Determinant of a Matrix

Determinant of a Matrix: is a special number that can be calculated from elements of a square matrix ( a matrix having equal no. of rows and columns).

The determinant of a square matrix A is denoted by det A or | A |.

Things to keep in mind: 

1.  Determinant only exists for a square matrix.

2.  The determinant can be a negative number.

For a 1 x 1  Matrix

For a  1 x 1  matrix ( 1 row and 1 column )=>

The determinant is |A| = a.

For a 2 x 2  Matrix

For a  2 x 2  matrix ( 2 rows and 2 columns )=>

The determinant is |A| = ad – bc.

For a 3 x 3 Matrix

For a  3 x 3  matrix ( 3 rows and 3 columns )=>

The determinant is |A| = a( ei – fh ) – b( di – gf ) + c( dh – eg ).

Cofactor of an element: is a number associated with an element in a square matrix, equal to the determinant of the matrix formed by removing the row and column in which the element appears from the given determinant.

Program to find determinant of a matrix in C++

#include<iostream>
#include<math.h>       // used for pow() function. pow() function is used to calculate some power of a number
using namespace std;

int find_determinant(int matrix[3][3], int n)
{
    int a,b,c,d,i,j,k;
    int temp_determinant=0,subcofactor_matrix[3][3],cofactor_i,cofactor_j;
    
    if(n==1)          // if order of a matrix is 1
        return matrix[0][0];
    
    if(n==2)         // if order of a matrix is 2
    {
       a=matrix[0][0];
       b=matrix[0][1];
       c=matrix[1][0];
       d=matrix[1][1];
       
       return (a*d - b*c);
    }
    
    if(n==3)        // if order of a matrix is 3
    {
        // loop for 0th row elements. Because for finding determinant of a matrix we only need to find out cofactors of 0th row elements
        for(k=0; k<n; k++) 
        {
            cofactor_i = 0;

                //Here, we have started loop from 1. Because we don't want to consider elements of same row for finding cofactor of an element
            for (i=1; i<n; i++) 
            {
               cofactor_j = 0;
               for (j=0; j<n; j++) 
                {
                    if (j!=k)  // we have used this codition. because we don't want to consider elements of same column for finding cofactor of an element
                    {
                        //forming sub_cofactor matrix
                        subcofactor_matrix[cofactor_i][cofactor_j] = matrix[i][j];
                        cofactor_j++;                        
                    }

                }
                cofactor_i++;
            }
            
            // calculating determinant of a matrix
            temp_determinant = temp_determinant + (pow(-1, k) * matrix[0][k] * find_determinant( subcofactor_matrix, n - 1 ));
        }
    }
    return temp_determinant;
}


int main()
{
    int i,j,n,determinant,matrix[3][3];
    
    cout<<"Enter order of a matrix (1/2/3): ";
    cin>>n;
    while( n<1 || n>3 )            // if mentioned order is not in between 1 and 3
    {
        cout<<"\nPlease..Enter order of a matrix in 1 to 3: ";
        cin>>n;
    }
    
    cout<<"\nEnter the elements of a matrix: ";       // Taking elements of matrix as a input
    for(i=0; i<n; i++)
        for(j=0; j<n; j++)
            cin>>matrix[i][j];
    
    cout<<"\nEntered matrix:\n";        // Displaying entered matrix
    for(i=0; i<n; i++)
    {
        for(j=0; j<n; j++)
        {
            cout<<matrix[i][j]<<" ";
        }
        cout<<"\n";
    }
            
    determinant=find_determinant(matrix,n);     // function call to find_determinant
    cout<<"\nDeterminant of the Entered matrix is: "<<determinant;
    return 0;
}

Input/Output:

Enter order of a matrix (1/2/3): 3                                                                                               
                                                                                                                               
Enter the elements of a matrix: 2 -3 1 2 0 -1 1 4 5                                                                                                
                                                                                                                              
Entered matrix:                                                                                                                  
2 -3 1                                                                                                                           
2 0 -1                                                                                                                           
1 4 5 
                                                                                                                           
Determinant of the Entered matrix is: 49