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# Determinant of a Matrix

• Difficulty Level : Medium
• Last Updated : 07 Aug, 2021

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:  ## C++

 // C++ program to find Determinant of a matrix#include 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;     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[f]        getCofactor(mat, temp, 0, f, n);        D += sign * mat[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++)            cout <<"  " <<  mat[i][j];        cout <<"n";    }} // Driver program to test above functionsint 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    cout <<"Determinant of the matrix is : " << determinantOfMatrix(mat, N);    return 0;} // this code is contributed by shivanisinghss2110

## C

 // C program to find Determinant of a matrix#include  // 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;     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[f]        getCofactor(mat, temp, 0, f, n);        D += sign * mat[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 functionsint 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;}

## Java

 // Java program to find Determinant of// a matrixclass 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;         // 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[f]            getCofactor(mat, temp, 0, f, n);            D += sign * mat[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.

## Python3

 # 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 * mat - mat * mat        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[current_column] * sub_det)     # returning the final Sum    return Sum  # Driver codeif __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.

## C#

 // C# program to find Determinant of// a matrixusing 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[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.

## Javascript

 
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,

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 properties of Determinant. In this approach we are converting the given matrix into upper triangular matrix using determinant properties The determinant of upper triangular 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:

## C++

 // C++ program to find Determinant of a matrix#include using namespace std; // Dimension of input square matrix#define N 4// Function to get determinant of matrixint 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 determinant 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);        }    }     // multiplying 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 codeint 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;}

## Java

 // Java program to find Determinant of a matrixclass 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 determinant 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);            }        }         // multiplying 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

## Python3

 # Python program to find Determinant of a matrix  def determinantOfMatrix(mat, n):     temp = *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 = i  # 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 determinant 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);     # multiplying 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 codeif __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))

## C#

 // C# program to find Determinant of a matrixusing 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 determinant 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);            }        }         // multiplying 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

## Javascript

 Javascript
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 argument.

## Python3

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

Output:

Determinant of the matrix is: 30.0

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