determinant.py

"""
Write a function that accepts a square matrix (N x N 2D array) and returns the determinant of the matrix.

How to take the determinant of a matrix -- it is simplest to start with the smallest cases:

A 1x1 matrix |a| has determinant a.

A 2x2 matrix [ [a, b], [c, d] ] or

|a  b|
|c  d|
has determinant: a*d - b*c.

The determinant of an n x n sized matrix is calculated by reducing the problem to the calculation of the determinants of n matrices ofn-1 x n-1 size.

For the 3x3 case, [ [a, b, c], [d, e, f], [g, h, i] ] or

|a b c|  
|d e f|  
|g h i|  
the determinant is: a * det(a_minor) - b * det(b_minor) + c * det(c_minor) where det(a_minor) refers to taking the determinant of the 2x2 matrix created by crossing out the row and column in which the element a occurs:

|- - -|
|- e f|
|- h i|  
Note the alternation of signs.

The determinant of larger matrices are calculated analogously, e.g. if M is a 4x4 matrix with first row [a, b, c, d], then:

det(M) = a * det(a_minor) - b * det(b_minor) + c * det(c_minor) - d * det(d_minor)
"""


def determinant(matrix):
    n = len(matrix)
    
    # Base case for a 1x1 matrix
    if n == 1:
        return matrix[0][0]
 
    
    # Base case for a 2x2 matrix
    if n == 2:
        return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0]
    
    det = 0
    
    for i in range(n):
        # Create the minor matrix by removing the current row and column
        minor = [row[:i] + row[i+1:] for row in matrix[1:]]

        # Calculate the determinant recursively using the minor matrix
        det += (-1) ** i * matrix[0][i] * determinant(minor)
    
    return det

# Example usage
print(determinant([[1, 2], [3, 4]]))