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:



 

Determinant of 3 x 3 Matrix: 

 

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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 } };
 
    // Function call
    printf("Determinant of the matrix is : %d",
           determinantOfMatrix(mat, N));
    return 0;
}
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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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# python program to find
# determinant of matrix.
 
# defining a function to get the
# minor matrix after excluding
# i-th row and j-th column.
 
 
def getcofactor(m, i, j):
    return [row[: j] + row[j+1:] for row in (m[: i] + m[i+1:])]
 
# defining the function to
# calculate determinant value
# of given matrix a.
 
 
def determinantOfMatrix(mat):
 
    # if given matrix is of order
    # 2*2 then simply return det
    # value by cross multiplying
    # elements of matrix.
    if(len(mat) == 2):
        value = mat[0][0] * mat[1][1] - mat[1][0] * mat[0][1]
        return value
 
    # initialize Sum to zero
    Sum = 0
 
    # loop to traverse each column
    # of matrix a.
    for current_column in range(len(mat)):
 
        # calculating the sign corresponding
        # to co-factor of that sub matrix.
        sign = (-1) ** (current_column)
 
        # calling the function recursily to
        # get determinant value of
        # sub matrix obtained.
        sub_det = determinantOfMatrix(getcofactor(mat, 0, current_column))
 
        # adding the calculated determinant
        # value of particular column
        # matrix to total Sum.
        Sum += (sign * mat[0][current_column] * sub_det)
 
    # returning the final Sum
    return Sum
 
 
# Driver code
if __name__ == '__main__':
 
    # declaring the matrix.
    mat = [[1, 0, 2, -1],
           [3, 0, 0, 5],
           [2, 1, 4, -3],
           [1, 0, 5, 0]]
 
    # printing determinant value
    # by function call
    print('Determinant of the matrix is :', determinantOfMatrix(mat))
 
# This code is contributed by Amit Mangal.
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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 
Below is the implementation of the above approach:

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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 } };
 
    // Function call
    printf("Determinant of the matrix is : %d",
           determinantOfMatrix(mat, N));
    return 0;
}
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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 determinant of matrix
    static int determinantOfMatrix(int mat[][], int n)
    {
        int num1, num2, det = 1, index,
                        total = 1; // Initialize result
 
        // temporary array for storing row
        int[] temp = new int[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, i, j);
                }
                // determinant sign changes when we shift
                // rows go through determinant properties
                det = (int)(det * Math.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);
    }
 
    static int[][] swap(int[][] arr, int i1, int j1, int i2,
                        int j2)
    {
        int temp = arr[i1][j1];
        arr[i1][j1] = arr[i2][j2];
        arr[i2][j2] = temp;
        return arr;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        /*int mat[N][N] = {{6, 1, 1},
                        {4, -2, 5},
                        {2, 8, 7}}; */
 
        int mat[][] = { { 1, 0, 2, -1 },
                        { 3, 0, 0, 5 },
                        { 2, 1, 4, -3 },
                        { 1, 0, 5, 0 } };
 
        // Function call
        System.out.printf(
            "Determinant of the matrix is : %d",
            determinantOfMatrix(mat, N));
    }
}
 
// This code is contributed by Rajput-Ji
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# Python program to find Determinant of a matrix
 
 
def determinantOfMatrix(mat, n):
 
    temp = [0]*# 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)
     
    # Function call
    print("Determinant of the matrix is : ", determinantOfMatrix(mat, N))
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// C# program to find Deteminant of a matrix
using System;
 
class GFG {
 
    // Dimension of input square matrix
    static readonly int N = 4;
 
    // Function to get determinant of matrix
    static int determinantOfMatrix(int[, ] mat, int n)
    {
        int num1, num2, det = 1, index,
                        total = 1; // Initialize result
 
        // temporary array for storing row
        int[] temp = new int[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, i, j);
                }
                // determinant sign changes when we shift
                // rows go through determinant properties
                det = (int)(det * Math.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);
    }
 
    static int[, ] swap(int[, ] arr, int i1, int j1, int i2,
                        int j2)
    {
        int temp = arr[i1, j1];
        arr[i1, j1] = arr[i2, j2];
        arr[i2, j2] = temp;
        return arr;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        /*int mat[N,N] = {{6, 1, 1},
                        {4, -2, 5},
                        {2, 8, 7}}; */
 
        int[, ] mat = { { 1, 0, 2, -1 },
                        { 3, 0, 0, 5 },
                        { 2, 1, 4, -3 },
                        { 1, 0, 5, 0 } };
 
        // Function call
        Console.Write("Determinant of the matrix is : {0}",
                      determinantOfMatrix(mat, N));
    }
}
 
// This code is contributed by 29AjayKumar
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Output
Determinant of the matrix is : 30


Time complexity : O(n3
Auxiliary Space : O(n) 
 

Method 3 : Using numpy package in python

There is a built in function or method in linalg module of numpy package in python. It can be called as numpy.linalg.det(mat) which returns the determinant value of matrix mat passed in the arguement.

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# importing the numpy package
# as np
import numpy as np
 
def determinant(mat):
     
    # calling the det() method
    det = np.linalg.det(mat)
    return round(det)
 
# Driver Code
# declaring the matrix
mat = [[1, 0, 2, -1],
       [3, 0, 0, 5],
       [2, 1, 4, -3],
       [1, 0, 5, 0]]
 
# Function call
print('Determinant of the matrix is:',
      determinant(mat))
 
# This code is contributed by Amit Mangal.
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Output:

Determinant of the matrix is: 30.0

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