# 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:

 // C++ program to find Deteminant 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[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; }

 // 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.

 # 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.

 // 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.

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:

 // C++ program to find Deteminant of a matrix #include  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; }

 // 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

 # 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 = 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 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))

 // 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

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.

 # 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.

Output:

Determinant of the matrix is: 30.0

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

Article Tags :
Practice Tags :