Step by Step solution about How to resolve the algorithm Cramer's rule step by step in the Ruby programming language

This Ruby code implements Cramer's rule, a method for solving a system of n linear equations with n unknowns. It takes a square matrix a and a column vector terms and returns the solution to the system of equations.

The code first checks if the input matrix a is square, meaning it has the same number of rows and columns. If it's not square, an ArgumentError is raised with the message "Matrix not square."

If the matrix is square, the code proceeds to calculate the solution. It does this by transposing the matrix a to obtain the columns of the matrix. Then, for each column in the transposed matrix, it creates a copy of the columns and replaces the column at the current index with the terms vector.

Next, it calculates the determinant of the resulting matrix and divides it by the determinant of the original matrix a. This gives the solution to the system of equations for the current unknown.

The code repeats this process for each column in the transposed matrix, and the result is an array of solutions for each unknown.

Finally, the code prints the array of solutions.

For example, given the matrix:

matrix = Matrix[
   [2, -1,  5,  1],
   [3,  2,  2, -6],
   [1,  3,  3, -1],
   [5, -2, -3,  3],
]

And the vector:

vector = [-3, -32, -47, 49]

The code would output the solution:

[-1.0, 11.0, 4.0, 1.0]

Which represents the values of x, y, z, and w that satisfy the system of equations:

2x - y + 5z + w = -3
3x + 2y + 2z - 6w = -32
x + 3y + 3z - w = -47
5x - 2y - 3z + 3w = 49

require 'matrix'
 
def cramers_rule(a, terms)
  raise ArgumentError, " Matrix not square"  unless a.square?
  cols = a.to_a.transpose
  cols.each_index.map do |i|
    c = cols.dup
    c[i] = terms
    Matrix.columns(c).det / a.det
  end
end

matrix = Matrix[
    [2, -1,  5,  1],
    [3,  2,  2, -6],
    [1,  3,  3, -1],
    [5, -2, -3,  3],
]

vector = [-3, -32, -47, 49]
puts cramers_rule(matrix, vector)