Matrix-Determinant

Matrix Determinant

see in codewars

Instructions

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 of n-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)

My solution

mathematical solution : Laplace expansion

Laplace expansion expresses the determinant of a matrix A in terms of determinants of smaller matrices, known as its minors.

The minor Mj is defined to be the determinant of the (n-1)*(n-1)-matrix that results from A by removing the 1st row and the j-th column. The expression is known as a cofactor.

a_{j} is the value of the j index in the 1st row

Javascript solution

Define Matrix class with :

• Attributes :

  • matrix : 2D array of values.

  • size : the size of the matrix.

  • minors : Array of minors matrices of the 1st row.

• Methodes :

  • minor() : to calculate the minors matrices of the 1st row.
  • det() : to calculate the determinant recursively with Laplace expansion formula