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)