Program to check idempotent matrix

Last Updated : 23 Dec, 2024

Given a square matrix mat[][] of order n*n, the task is to check if it is an Idempotent Matrix or not.

Idempotent matrix: A matrix is said to be an idempotent matrix if the matrix multiplied by itself returns the same matrix, i.e. the matrix mat[][] is said to be an idempotent matrix if and only if M * M = M.

Examples:

Input: mat[][] = [[2, -2, -4],
[-1, 3, 4],
[1, -2, 3]]
Output: True
Explanation: Given matrix if multiplied by itself returns the same matrix.

Input: mat[][] = [[1, 2],
[3, 4]]
Output: False
Explanation: Given matrix if multiplied by itself returns the different matrix.

Approach:

The idea is to multiply matrix mat[][] by itself and check if all the elements of resultant matrix res[][] is equal to corresponding elements of mat[][]. Firstly multiply matrix mat[][] by itself using Matrix Multiplication and store the result in matrix res[][]. Then compare if both the matrices are equal or not.

C++

// Program to check given matrix  // is idempotent matrix or not. #include<bits/stdc++.h> using namespace std;  // Function for matrix multiplication. void mulMatrix(vector<vector<int>>& mat,               vector<vector<int>>& res) {     int n = mat.size();        for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++) {             for (int k = 0; k < n; k++) {                 res[i][j] += mat[i][k] * mat[k][j];             }         }     } }  // Function to check idempotent // property of matrix. bool checkIdempotent(vector<vector<int>>& mat) {        int n = mat.size();      // multiply matrix mat by iself     // and store it into res.     vector<vector<int>> res(n, vector<int>(n, 0));     mulMatrix(mat, res);      // check if all the elements of      // res are same as mat.     for (int i = 0; i < n; i++)             for (int j = 0; j < n; j++)                     if (mat[i][j] != res[i][j])                 return false;     return true; }  int main() {     vector<vector<int>> mat = {{2, -2, -4},                     {-1, 3, 4},                     {1, -2, -3}};          if (checkIdempotent(mat))         cout << "True";     else         cout << "False";     return 0; }

Java

// Java program to check given matrix  // is idempotent matrix or not. import java.io.*;  class GfG {          // Function for matrix multiplication.     static void mulMatrix(int[][] mat, int[][] res) {         int n = mat.length;          // Perform matrix multiplication         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {                 res[i][j] = 0;                 for (int k = 0; k < n; k++) {                     res[i][j] += mat[i][k] * mat[k][j];                 }             }         }     }          // Function to check idempotent     // property of matrix.     static boolean checkIdempotent(int mat[][]) {          int n = mat.length;          // Calculate multiplication of matrix         // with itself and store it into res.         int res[][] = new int[n][n];         mulMatrix(mat, res);              for (int i = 0; i < n; i++) {              for (int j = 0; j < n; j++) {                 if (mat[i][j] != res[i][j])                     return false;             }         }         return true;     }      public static void main (String[] args) {         int mat[][] = {{2, -2, -4},                        {-1, 3, 4},                        {1, -2, -3}};              if (checkIdempotent(mat))             System.out.println("True");         else             System.out.println("False");              } }

Python

# Python Program to check given matrix  # is idempotent matrix or not. import math  # Function for matrix multiplication. def mulMatrix(mat, res):     n = len(mat)      # Perform matrix multiplication     for i in range(n):         for j in range(n):             for k in range(n):                 res[i][j] += mat[i][k] * mat[k][j]  # Function to check idempotent # property of matrix. def checkIdempotent(mat):     n = len(mat)      # Calculate multiplication of matrix     # with itself and store it into res.     res =[[0]*n for i in range(0,n)]     mulMatrix(mat, res)      for i in range(0,n):         for j in range(0,n):                  if (mat[i][j] != res[i][j]):                 return False     return True  mat = [ [2, -2, -4],         [-1, 3, 4],         [1, -2, -3] ]  if (checkIdempotent(mat)):     print("True") else:     print("False")

// C# program to check given matrix  // is idempotent matrix or not. using System;  class GfG {      // Function for matrix multiplication.     static void mulMatrix(int[,] mat, int[,] res) {         int n = mat.GetLength(0);          // Perform matrix multiplication         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {               	res[i, j] = 0;                 for (int k = 0; k < n; k++) {                     res[i, j] += mat[i, k] * mat[k, j];                 }             }         }     }          // Function to check idempotent     // property of matrix.     static bool checkIdempotent(int[,] mat){          int n = mat.GetLength(0);          // Calculate multiplication of matrix         // with itself and store it into res.         int [,]res = new int[n,n];         mulMatrix(mat, res);              for (int i = 0; i < n; i++) {              for (int j = 0; j < n; j++) {                 if (mat[i,j] != res[i,j])                     return false;             }         }         return true;     }      static void Main () {         int [,]mat = {{2, -2, -4},                     {-1, 3, 4},                     {1, -2, -3}};              if (checkIdempotent(mat))             Console.WriteLine( "True");         else             Console.WriteLine("False");              } }

JavaScript

// Javascript program to check given matrix  // is idempotent matrix or not.  // Function for matrix multiplication. function mulMatrix(mat, res) {     const n = mat.length;      // Perform matrix multiplication     for (let i = 0; i < n; i++) {         for (let j = 0; j < n; j++) {             for (let k = 0; k < n; k++) {                 res[i][j] += mat[i][k] * mat[k][j];             }         }     } }  // Function to check idempotent // property of matrix. function checkIdempotent(mat) {       const n = mat.length;      // Calculate multiplication of matrix     // with itself and store it into res.     const res = Array.from(Array(n), ()=>Array(n).fill(0));     mulMatrix(mat, res);      for (let i = 0; i < n; i++) {          for (let j = 0; j < n; j++) {             if (mat[i][j] != res[i][j])                 return false;         }     }     return true; }  //driver code const mat = [[2, -2, -4],             [-1, 3, 4],             [1, -2, -3]];              if (checkIdempotent(mat))     console.log( "True"); else     console.log("False");

Output

True

Time Complexity: O(n³)
Auxiliary Space: O(n²), since n²extra space has been taken to store the result of multiplication.

Program to check diagonal matrix and scalar matrix

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Program to check idempotent matrix

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