The determinant of a Matrix is defined as a special number that is defined only for square matrices (matrices that have the same number of rows and columns). A determinant is used in many places in calculus and other matrices related to algebra, it actually represents the matrix in terms of a real number which can be used in solving a system of a linear equation and finding the inverse of a matrix.
Determinant of 2 x 2 Matrix:
Determinant of 2 x 2 matrix
Determinant of 3 x 3 Matrix:
Determinant of 3 x 3 matrix
How to calculate?
The value of the determinant of a matrix can be calculated by the following procedure:
- For each element of the first row or first column get the cofactor of those elements.
- Then multiply the element with the determinant of the corresponding cofactor.
- Finally, add them with alternate signs. As a base case, the value of the determinant of a 1*1 matrix is the single value itself.
The cofactor of an element is a matrix that we can get by removing the row and column of that element from that matrix.
Code block
Output
Determinant of the matrix is : 30
Time Complexity: O(n4)
Space Complexity: O(n2), Auxiliary space used for storing cofactors.
Note: In the above recursive approach when the size of the matrix is large it consumes more stack size.
Determinant of a Matrix using Determinant properties:
- In this method, we are using the properties of Determinant.
- Converting the given matrix into an upper triangular matrix using determinant properties
- The determinant of the upper triangular matrix is the product of all diagonal elements.
- Iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties
- If the diagonal element is zero then search for the 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 a matrix is zero
- Case 2: If there exists a non-zero element there exist two cases
- Case A: If the index is with a respective diagonal row element. Using the determinant properties make all the column elements down to it zero
- Case B: Swap the row with respect to the 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 <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 (index < n && mat[index][i] == 0) { index++; } if (index == n) // if there is non zero element { // the determinant of matrix as zero continue; } if (index != i) { // loop for swapping 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 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
// Java program to find Determinant 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 (index < n && mat[index][i] == 0) { index++; } if (index == n) // if there is non zero element { // the determinant of matrix as zero continue; } if (index != i) { // loop for swapping 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 = [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(index < n and mat[index][i] == 0): index += 1 if(index == n): # if there is non zero element # the determinant of matrix as zero continue if(index != i): # loop for swapping 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 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#
// C# program to find Determinant 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(index < n && mat[index, i] == 0) { index++; } if (index == n) // if there is non zero element { // the determinant of matrix as zero continue; } if (index != i) { // loop for swapping 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<script> // javascript program to find Determinant of a matrix // Dimension of input square matrix var N = 4; // Function to get determinant of matrix function determinantOfMatrix(mat , n) { var num1, num2, det = 1, index, total = 1; // Initialize result // temporary array for storing row var temp = Array(n + 1).fill(0); // loop for traversing the diagonal elements for (i = 0; i < n; i++) { index = i; // initialize the index // finding the index which has non zero value while (index < n && mat[index][i] == 0) { index++; } if (index == n) // if there is non zero element { // the determinant of matrix as zero continue; } if (index != i) { // loop for swapping the diagonal element row // and index row for (j = 0; j < n; j++) { swap(mat, index, j, i, j); } // determinant sign changes when we shift // rows go through determinant properties det = parseInt((det * Math.pow(-1, index - i))); } // storing the values of diagonal row elements for (j = 0; j < n; j++) { temp[j] = mat[i][j]; } // traversing every row below the diagonal // element for (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 (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 (i = 0; i < n; i++) { det = det * mat[i][i]; } return (det / total); // Det(kA)/k=Det(A); } function swap(arr , i1 , j1 , i2, j2) { var temp = arr[i1][j1]; arr[i1][j1] = arr[i2][j2]; arr[i2][j2] = temp; return arr; } // Driver code /*var mat[N][N] = [{6, 1, 1], {4, -2, 5], {2, 8, 7}]; */ var mat = [ [ 1, 0, 2, -1 ], [ 3, 0, 0, 5 ], [ 2, 1, 4, -3 ], [ 1, 0, 5, 0 ] ]; // Function call document.write( "Determinant of the matrix is : ", determinantOfMatrix(mat, N)); // This code contributed by gauravrajput1 </script> |
Output
Determinant of the matrix is : 30
Time complexity: O(n3)
Auxiliary Space: O(n), Space used for storing row.
Determinant of a Matrix Using the NumPy package in Python
There is a built-in function or method in linalg module of NumPy package in python. It can be called numpy.linalg.det(mat) which returns the determinant value of the matrix mat passed in the argument.
Python3
# 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)) |
Output:
Determinant of the matrix is: 30
Time Complexity: O(n3), as the time complexity of np.linalg.det is O(n3) for an n x n order matrix.
Auxiliary Space: O(1)
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