Computing the determinant is one of the fundamental operations in linear algebra with many practical applications. The determinant can be used to compute the area of a parallelogram or to check if there exists a solution for a system of linear equations. The determinant is defined only for square matrices and can be computed recursively.

If \(\alpha_{ij}\) is the element of the matrix A at row i and column j with \(i,j \geq 1\) then the determinant can be computed as follows:

where \(\tilde{A}_{ij}\) is the matrix A with row i and column j being removed. The recursion terminates if the matrix A is a 1x1 matrix because the determinant of a 1x1 matrix simply is \(\alpha_{11}\).

Translating this into Python code we finally get:

import numpy as np

def det(A):
    return sum([(-1)**i * A[i, 0] * det(np.delete(np.delete(A, 0, 1), i, 0)) for i in range(A.shape[0])]) if A.shape != (1, 1) else A[0, 0]

m = np.array([[1, 4], [2, 1]])
print det(m)

Note: this is not a very efficient way to compute the determinant of a matrix because this approach takes O(n!) steps which is very inefficient even for rather small matrices. If you really need an efficient way which returns the solution in a finite time even for big matrices you should use numpy.linalg.det from numpy instead.