Matrix Determinant

Matrix Determinant

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)

Solutions

🐍 Python

def subdet(m):
    det = 0
    if len(m) != 2:
        for n,i in enumerate(range(len(m))):
            if n%2:
                det -= m[0][i] * subdet([ mi[:i]+mi[i+1:] for mi in m[1:] ])
            else:
                det += m[0][i] * subdet([ mi[:i]+mi[i+1:] for mi in m[1:] ])
    else:
        det = (m[0][0]*m[1][1]-m[0][1]*m[1][0])

    return det

def determinant(m):
    sz = len(m)

    if sz == 1:
      return (m[0][0])
    else:
        return subdet(m)