Python Program to check Involutory Matrix

Given a matrix and the task is to check matrix is an involutory matrix or not. 
Involutory Matrix: A matrix is said to be an involutory matrix if the matrix multiplies by itself and returns the identity matrix. The involutory matrix is the matrix that is its own inverse. The matrix A is said to be an involutory matrix if A * A = I. Where I is the identity matrix. 

Examples:

Input : mat[N][N] = {{1, 0, 0},
                     {0, -1, 0},
                     {0, 0, -1}}
Output : Involutory Matrix

Input : mat[N][N] = {{1, 0, 0},
                     {0, 1, 0},
                     {0, 0, 1}} 
Output : Involutory Matrix

Python3

# Program to implement involutory matrix.
N = 3;

# Function for matrix multiplication.
def multiply(mat, res):

    for i in range(N):
        for j in range(N):
            res[i][j] = 0;
            for k in range(N):
                res[i][j] += mat[i][k] * mat[k][j];
    return res;

# Function to check involutory matrix.
def InvolutoryMatrix(mat):

    res=[[0 for i in range(N)]
            for j in range(N)];

    # multiply function call.
    res = multiply(mat, res);

    for i in range(N):
        for j in range(N):
            if (i == j and res[i][j] != 1):
                return False;
            if (i != j and res[i][j] != 0):
                return False;
    return True;

# Driver Code
mat = [[1, 0, 0], [0, -1, 0], [0, 0, -1]];

# Function call. If function
# return true then if part
# will execute otherwise
# else part will execute.
if (InvolutoryMatrix(mat)):
    print("Involutory Matrix");
else:
    print("Not Involutory Matrix");

# This code is contributed by mits

Output

Involutory Matrix

Time Complexity: O(N³)
Auxiliary Space: O(N²)

Please refer complete article on Program to check Involutory Matrix for more details!

Using Numpy:

Note: Before running the code please install the Numpy library using the command below

pip install numpy

Another approach  is to use the numpy library to check if the matrix is involutory. This can be done by using the numpy.allclose function to compare the matrix with its inverse.

For example, the following code snippet checks if a matrix is involutory:

Python3

import numpy as np

def is_involutory(matrix):
    # Calculate the inverse of the matrix using the numpy linalg module
    inverse = np.linalg.inv(matrix)

    # Check if the matrix is equal to its inverse using numpy.allclose
    return np.allclose(matrix, inverse)

# Example usage
matrix = np.array([[1, 0, 0], [0, -1, 0], [0, 0, -1]])
print(is_involutory(matrix)) # prints True
#This code is contributed by Edula Vinay Kumar Reddy

Output: True

This approach has the advantage of being more concise and easier to read, and it also takes advantage of the optimized linear algebra routines provided by numpy. Space complexity is O(N^2) and the time complexity of this approach will depend on the complexity of the matrix inverse calculation, which is generally O(N^3) for dense matrices.