A matrix is said to be singular if the determinant of the matrix is 0 otherwise it is non-singular .
Examples:
Input : 0 0 0
4 5 6
1 2 3
Output : Yes
Determinant value of the matrix is
0 (Note first row is 0)
Input : 1 0 0
4 5 6
1 2 3
Output : No
Determinant value of the matrix is 3
(which is non-zero).
First find the determinant of the matrix and the check the condition if the determinant id 0 or not, if it is 0 then matrix is a singular matrix otherwise it is a non-singular matrix .
Implementation:
C++
// C++ program check if a matrix is// singular or not.#include <bits/stdc++.h>using namespace std;#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 to check if mat[][] is singular or not. */bool isSingular(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] * isSingular(temp, n - 1); // terms are to be added with alternate sign sign = -sign; } return D;}// Driver program to test above functionsint main(){ int mat[N][N] = { { 4, 10, 1 }, { 0, 0, 0 }, { 1, 4, -3 } }; if (isSingular(mat, N)) cout << "Matrix is Singular" << endl; else cout << "Matrix is non-singular" << endl; return 0;} |
Java
// Java program check if a matrix is // singular or not. class GFG { static final int N = 3; // 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 to check if mat[][] is singular or not. */ static int isSingular(int mat[][], int n) { int D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) { return mat[0][0]; } int temp[][] = new int[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] * isSingular(temp, n - 1); // terms are to be added with alternate sign sign = -sign; } return D; } // Driver code public static void main(String[] args) { int mat[][] = {{4, 10, 1}, {0, 0, 0}, {1, 4, -3}}; if (isSingular(mat, N) == 1) { System.out.println("Matrix is Singular"); } else { System.out.println("Matrix is non-singular"); } }}/* This code contributed by PrinciRaj1992 */ |
Python3
# python 3 program check if a matrix is# singular or not.global NN = 3# Function to get cofactor of mat[p][q] in temp[][].# n is current dimension of mat[][]def getCofactor(mat,temp,p,q,n): i = 0 j = 0 # Looping for each element of the matrix for row in range(n): for col in range(n): # Copying into temporary matrix only # those element which are not in given # row and column if (row != p and col != q): temp[i][j] = mat[row][col] j += 1 # Row is filled, so increase row # index and reset col index if (j == n - 1): j = 0 i += 1# Recursive function to check if mat[][] is# singular or not. */def isSingular(mat,n): D = 0 # Initialize result # Base case : if matrix contains single element if (n == 1): return mat[0][0] temp = [[0 for i in range(N + 1)] for i in range(N + 1)]# To store cofactors sign = 1 # To store sign multiplier # Iterate for each element of first row for f in range(n): # Getting Cofactor of mat[0][f] getCofactor(mat, temp, 0, f, n) D += sign * mat[0][f] * isSingular(temp, n - 1) # terms are to be added with alternate sign sign = -sign return D# Driver program to test above functionsif __name__ == '__main__': mat = [[4, 10, 1],[0, 0, 0],[1, 4, -3]] if (isSingular(mat, N)): print("Matrix is Singular") else: print("Matrix is non-singular")# This code is contributed by# Surendra_Gangwar |
C#
// C# program check if a matrix is // singular or not.using System;class GFG { static readonly int N = 3; // 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 to check if mat[,] is singular or not. */ static int isSingular(int [,]mat, int n) { int D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) { return mat[0, 0]; } int [,]temp = new int[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] * isSingular(temp, n - 1); // terms are to be added with alternate sign sign = -sign; } return D; } // Driver code public static void Main(String[] args) { int [,]mat = {{4, 10, 1}, {0, 0, 0}, {1, 4, -3}}; if (isSingular(mat, N) == 1) { Console.WriteLine("Matrix is Singular"); } else { Console.WriteLine("Matrix is non-singular"); } }}// This code contributed by Rajput-Ji |
Javascript
<script> // Javascript program check if a matrix is // singular or not.var N = 3;// Function to get cofactor of mat[p,q] in temp[,]. // n is current dimension of mat[,] function getCofactor(mat, temp, p, q, n){ var i = 0, j = 0; // Looping for each element of the matrix for (var row = 0; row < n; row++) { for (var 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 to check if mat[,] is singular or not. */function isSingular(mat, n) { var D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) { return mat[0][0]; } var temp = Array.from(Array(N), ()=>Array(N));// To store cofactors var sign = 1; // To store sign multiplier // Iterate for each element of first row for(var f = 0; f < n; f++) { // Getting Cofactor of mat[0,f] getCofactor(mat, temp, 0, f, n); D += sign * mat[0][f] * isSingular(temp, n - 1); // terms are to be added with alternate sign sign = -sign; } return D;}// Driver code var mat = [[4, 10, 1], [0, 0, 0], [1, 4, -3]];if (isSingular(mat, N) == 1) { document.write("Matrix is Singular");} else{ document.write("Matrix is non-singular");}// This code is contributed by noob2000.</script> |
Output:
Matrix is non-singular
Time complexity: O(n3)
Auxiliary space: O(n2), for temp array to store co-factors
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