# 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.

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