Determinant of a Matrix

What is Determinant of a Matrix?
Determinant of a Matrix is a special number that is defined only for square matrices (matrices which have same number of rows and columns). Determinant is used at many places in calculus and other matrix related algebra, it actually represents the matrix in term of a real number which can be used in solving system of linear equation and finding the inverse of a matrix.

How to calculate?
The value of determinant of a matrix can be calculated by following procedure –
For each element of first row or first column get cofactor of those elements and then multiply the element with the determinant of the corresponding cofactor, and finally add them with alternate signs. As a base case the value of determinant of a 1*1 matrix is the single value itself.
Cofactor of an element, is a matrix which we can get by removing row and column of that element from that matrix.

Determinant of 2 x 2 Matrix:

 A = \begin{bmatrix} a & b\\  c & d \end{bmatrix}  \begin{vmatrix} A \end{vmatrix}= ad - bc

22

Determinant of 3 x 3 Matrix:
 A = \begin{bmatrix} a & b & c\\  d & e & f\\  g & h & i \end{bmatrix}  \begin{vmatrix} A \end{vmatrix}= a(ei-fh)-b(di-gf)+c(dh-eg)

C++

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// C++ program to find Deteminant of a matrix
#include <bits/stdc++.h>
using namespace std;
  
// Dimension of input square matrix
#define N 4
  
// Function to get cofactor of mat[p][q] in temp[][]. n is current
// dimension of mat[][]
void getCofactor(int mat[N][N], int temp[N][N], int p, int q, int n)
{
    int i = 0, j = 0;
  
    // Looping for each element of the matrix
    for (int row = 0; row < n; row++)
    {
        for (int col = 0; col < n; col++)
        {
            //  Copying into temporary matrix only those element
            //  which are not in given row and column
            if (row != p && col != q)
            {
                temp[i][j++] = mat[row][col];
  
                // Row is filled, so increase row index and
                // reset col index
                if (j == n - 1)
                {
                    j = 0;
                    i++;
                }
            }
        }
    }
}
  
/* Recursive function for finding determinant of matrix.
   n is current dimension of mat[][]. */
int determinantOfMatrix(int mat[N][N], int n)
{
    int D = 0; // Initialize result
  
    //  Base case : if matrix contains single element
    if (n == 1)
        return mat[0][0];
  
    int temp[N][N]; // To store cofactors
  
    int sign = 1;  // To store sign multiplier
  
     // Iterate for each element of first row
    for (int f = 0; f < n; f++)
    {
        // Getting Cofactor of mat[0][f]
        getCofactor(mat, temp, 0, f, n);
        D += sign * mat[0][f] * determinantOfMatrix(temp, n - 1);
  
        // terms are to be added with alternate sign
        sign = -sign;
    }
  
    return D;
}
  
/* function for displaying the matrix */
void display(int mat[N][N], int row, int col)
{
    for (int i = 0; i < row; i++)
    {
        for (int j = 0; j < col; j++)
            printf("  %d", mat[i][j]);
        printf("n");
    }
}
  
// Driver program to test above functions
int main()
{
    /* int mat[N][N] = {{6, 1, 1},
                     {4, -2, 5},
                     {2, 8, 7}}; */
  
    int mat[N][N] = {{1, 0, 2, -1},
                     {3, 0, 0, 5},
                     {2, 1, 4, -3},
                     {1, 0, 5, 0}
                    };
  
    printf("Determinant of the matrix is : %d",
            determinantOfMatrix(mat, N));
    return 0;
}

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Java

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// Java program to find Deteminant of 
// a matrix
class GFG {
      
    // Dimension of input square matrix
    static final int N = 4;
      
    // Function to get cofactor of 
    // mat[p][q] in temp[][]. n is 
    // current dimension of mat[][]
    static void getCofactor(int mat[][], 
             int temp[][], int p, int q, int n)
    {
        int i = 0, j = 0;
      
        // Looping for each element of 
        // the matrix
        for (int row = 0; row < n; row++)
        {
            for (int col = 0; col < n; col++)
            {
                  
                // Copying into temporary matrix 
                // only those element which are 
                // not in given row and column
                if (row != p && col != q)
                {
                    temp[i][j++] = mat[row][col];
      
                    // Row is filled, so increase 
                    // row index and reset col 
                    //index
                    if (j == n - 1)
                    {
                        j = 0;
                        i++;
                    }
                }
            }
        }
    }
      
    /* Recursive function for finding determinant
    of matrix. n is current dimension of mat[][]. */
    static int determinantOfMatrix(int mat[][], int n)
    {
        int D = 0; // Initialize result
      
        // Base case : if matrix contains single
        // element
        if (n == 1)
            return mat[0][0];
          
        // To store cofactors
        int temp[][] = new int[N][N]; 
          
        // To store sign multiplier
        int sign = 1
      
        // Iterate for each element of first row
        for (int f = 0; f < n; f++)
        {
            // Getting Cofactor of mat[0][f]
            getCofactor(mat, temp, 0, f, n);
            D += sign * mat[0][f] 
               * determinantOfMatrix(temp, n - 1);
      
            // terms are to be added with 
            // alternate sign
            sign = -sign;
        }
      
        return D;
    }
      
    /* function for displaying the matrix */
    static void display(int mat[][], int row, int col)
    {
        for (int i = 0; i < row; i++)
        {
            for (int j = 0; j < col; j++)
                System.out.print(mat[i][j]);
                  
            System.out.print("\n");
        }
    }
      
    // Driver code
    public static void main (String[] args)
    {
          
        int mat[][] = {{1, 0, 2, -1},
                        {3, 0, 0, 5},
                        {2, 1, 4, -3},
                        {1, 0, 5, 0}};
      
        System.out.print("Determinant "
                    "of the matrix is : "+
             determinantOfMatrix(mat, N));
    }
}
  
// This code is contributed by Anant Agarwal.

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C#

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// C# program to find Deteminant of 
// a matrix
using System;
class GFG {
      
    // Dimension of input square matrix
    static int N = 4;
      
    // Function to get cofactor of 
    // mat[p][q] in temp[][]. n is 
    // current dimension of mat[][]
    static void getCofactor(int [,]mat, 
                            int [,]temp, 
                            int p, 
                            int q,
                            int n)
    {
        int i = 0, j = 0;
      
        // Looping for each element of 
        // the matrix
        for (int row = 0; row < n; row++)
        {
            for (int col = 0; col < n; col++)
            {
                  
                // Copying into temporary matrix 
                // only those element which are 
                // not in given row and column
                if (row != p && col != q)
                {
                    temp[i, j++] = mat[row, col];
      
                    // Row is filled, so increase 
                    // row index and reset col 
                    //index
                    if (j == n - 1)
                    {
                        j = 0;
                        i++;
                    }
                }
            }
        }
    }
      
    /* Recursive function for
       finding determinant
       of matrix. n is current 
       dimension of mat[][]. */
    static int determinantOfMatrix(int [,]mat, 
                                   int n)
    {
        int D = 0; // Initialize result
      
        // Base case : if matrix 
        // contains single
        // element
        if (n == 1)
            return mat[0, 0];
          
        // To store cofactors
        int [,]temp = new int[N, N]; 
          
        // To store sign multiplier
        int sign = 1; 
      
        // Iterate for each element
        // of first row
        for (int f = 0; f < n; f++)
        {
              
            // Getting Cofactor of mat[0][f]
            getCofactor(mat, temp, 0, f, n);
            D += sign * mat[0, f] 
            * determinantOfMatrix(temp, n - 1);
      
            // terms are to be added with 
            // alternate sign
            sign = -sign;
        }
      
        return D;
    }
      
    /* function for displaying
       the matrix */
    static void display(int [,]mat, 
                        int row, 
                        int col)
    {
        for (int i = 0; i < row; i++)
        {
            for (int j = 0; j < col; j++)
                Console.Write(mat[i,j]);
                  
            Console.Write("\n");
        }
    }
      
    // Driver code
    public static void Main ()
    {
          
        int [,]mat = {{1, 0, 2, -1},
                      {3, 0, 0, 5},
                      {2, 1, 4, -3},
                      {1, 0, 5, 0}};
  
        Console.Write("Determinant "
                    "of the matrix is : "+
                     determinantOfMatrix(mat, N));
    }
}
  
// This code is contributed by nitin mittal.

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Output:

Determinant of the matrix is : 30

Adjoint and Inverse of a Matrix
There are various properties of the Determinant which can be helpful for solving problems related with matrices,

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

In Above Method Recursive Approach is discussed.When the size of matrix is large it consumes more stack size
In this Method We are using the properities of Determinant.In this approach we are converting the given matrix into upper triangular matrix using determinant properties The determinant of upper traingular matrix is the product of all diagonal elements For properties on determinant go through this website https://cran.r-project.org/web/packages/matlib/vignettes/det-ex1.html
In this approach we are iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties
If the diagonal element is zero then we will search next non zero element in the same column
There exist two cases
Case 1:
If there is no non zero element.In this case the determinant of matrix is zero
Case 2:
If there exists non zero element there exist two cases
Case a:
if index is with respective diagonal row element.Using the determinant properties we make all the column elements down to it as zero
Case b:
Here we need to swap the row with respective to diagonal element column and continue the case ‘a; operation
Time complexity : O(n^3)
Auxiliary Space : O(n)

C++

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// C++ program to find Deteminant of a matrix 
#include<bits/stdc++.h>  
using namespace std;  
    
// Dimension of input square matrix  
#define N 4  
// Function to get determinant of matrix  
int determinantOfMatrix(int mat[N][N], int n)  
{  
    int num1,num2,det = 1,index,total = 1; // Initialize result  
        
    // temporary array for storing row  
    int temp[n + 1];  
        
    //loop for traversing the diagonal elements 
    for(int i = 0; i < n; i++)   
    
        index = i; // initialize the index  
            
        //finding the index which has non zero value  
        while(mat[index][i] == 0 && index < n) {  
            index++;      
                
        }  
        if(index == n) // if there is non zero element  
        {  
            // the determinat of matrix as zero  
            continue;  
                
        }  
        if(index != i)  
        {  
            //loop for swaping the diagonal element row and index row  
            for(int j = 0; j < n; j++)  
            {  
                swap(mat[index][j],mat[i][j]);     
            }  
                //determinant sign changes when we shift rows  
                //go through determinant properties  
                det = det*pow(-1,index-i);    
       }  
           
       //storing the values of diagonal row elements  
       for(int j = 0; j < n; j++)  
       {  
           temp[j] = mat[i][j];  
               
       }  
       //traversing every row below the diagonal element  
       for(int j = i+1; j < n; j++)  
       {  
           num1 = temp[i]; //value of diagonal element  
           num2 = mat[j][i]; //value of next row element  
               
           //traversing every column of row  
           // and multiplying to every row  
           for(int k = 0; k < n; k++)  
           {  
               //multiplying to make the diagonal  
               // element and next row element equal  
               mat[j][k] = (num1 * mat[j][k]) - (num2 * temp[k]);  
                   
           }  
           total = total * num1; // Det(kA)=kDet(A);  
           }  
            
    }  
    
    //mulitplying the diagonal elements to get determinant  
    for(int i = 0; i < n; i++)  
    {  
        det = det * mat[i][i];  
            
    
    return (det/total); //Det(kA)/k=Det(A);  
    }  
    
// Driver code 
int main()  
{  
/*int mat[N][N] = {{6, 1, 1},  
                    {4, -2, 5},  
                    {2, 8, 7}}; */
    
    int mat[N][N] = {{1, 0, 2, -1},  
                    {3, 0, 0, 5},  
                    {2, 1, 4, -3},  
                    {1, 0, 5, 0}  
                    };  
        
    printf("Determinant of the matrix is : %d",  
            determinantOfMatrix(mat, N));  
    return 0;  
        
}  

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Python3

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#Python program to find Determinant of a matrix
  
def determinantOfMatrix(mat,n):
  
    temp = [0]*n # temporary array for storing row  
    total=1 
    det=1  # initialize result
  
    # loop for traversing the diagonal elements 
    for i in range(0,n):
        index=# initialize the index
  
        # finding the index which has non zero value  
        while(mat[index][i] == 0 and index < n):
            index+=1
     
        if(index == n): # if there is non zero element
            # the determinat of matrix as zero
            continue
  
        if(index != i):
            # loop for swaping the diagonal element row and index row
            for j in range(0,n):
                mat[index][j],mat[i][j] = mat[i][j],mat[index][j]
                  
            # determinant sign changes when we shift rows  
            # go through determinant properties  
            det = det*int(pow(-1,index-i))
    
        # storing the values of diagonal row elements 
        for j in range(0,n):
            temp[j] = mat[i][j]
           
        #traversing every row below the diagonal element
        for j in range(i+1,n):
            num1 = temp[i]     # value of diagonal element
            num2 = mat[j][i]   # value of next row element
  
            # traversing every column of row  
            # and multiplying to every row  
            for k in range(0,n):
                # multiplying to make the diagonal  
                # element and next row element equal
  
                mat[j][k] = (num1*mat[j][k]) - (num2*temp[k])
  
            total = total * num1 # Det(kA)=kDet(A);
  
    #mulitplying the diagonal elements to get determinant
    for i in range(0,n):
         det = det*mat[i][i]
  
  
    return int(det/total) #Det(kA)/k=Det(A);  
  
#Drivers code
if __name__ == "__main__":
    # mat=[[6 1 1][4 -2 5][2 8 7]]
  
    mat=[[1, 0, 2, -1],[3, 0, 0, 5],[2, 1, 4, -3],[1, 0, 5, 0]]
    N=len(mat)
    print("Determinant of the matrix is : ",determinantOfMatrix(mat,N))

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Output:

Determinant of the matrix is : 30


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