Solve Systems with Matrices

system_solution(matrix_one, matrix_two, precision=4)

Solves a system of equations using matrices (independent matrix multiplied by resultant matrix equals dependent matrix)

Parameters

  • matrix_one (list of lists of int or float) – List of lists of numbers representing the independent matrix of a system of equations

  • matrix_two (list of lists of int or float) – List of lists of numbers representing the dependent matrix of a system of equations

  • precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the results

Raises

  • TypeError – First and second arguments must be 2-dimensional lists

  • TypeError – Elements nested within first and second arguments must be integers or floats

  • ValueError – First and second arguments must contain the same amount of lists

  • ValueError – Last argument must be a positive integer

Returns

solution – Row vector of coefficients that if expressed as a column vector would satisfy the equation

Return type

list of float

See also

matrix_product(), transposed_matrix(), linear_determinant(), inverse_matrix()

Notes

  • Independent matrix: \(\mathbf{A} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}\)

  • Dependent matrix: \(\mathbf{B} = \begin{bmatrix} b_{1,1} \\ b_{2,1} \\ \cdots \\ b_{m,1} \end{bmatrix}\)

  • Variable matrix: \(\mathbf{X} = \begin{bmatrix} x_{1,1} \\ x_{2,1} \\ \cdots \\ x_{m,1} \end{bmatrix}\)

  • System of equations in terms of matrices: \(\mathbf{A}\cdot{\mathbf{X}} = \mathbf{B}\)

  • Solution of system of equations: \(\mathbf{X} = \mathbf{A}^{-1}\cdot{\mathbf{B}}\)

  • Solving a System of Equations Using Matrices

Examples

Import system_solution function from regressions library
>>> from regressions.matrices.solve import system_solution
Solve the system that has an independent matrix of [[2, 3], [1, -1]] and a dependent matrix of [[5], [1]]
>>> solution_2values = system_solution([[2, 3], [1, -1]], [[5], [1]])
>>> print(solution_2values)
[1.6, 0.6]
Solve the system that has an independent matrix of [[1, -2, 3], [-4, 5, -6], [7, -8, 9], [-10, 11, 12]] and a dependent matrix of [[2], [-3], [5], [-7]]
>>> solution_3values = system_solution([[1, -2, 3], [-4, 5, -6], [7, -8, 9], [-10, 11, 12]], [[2], [-3], [5], [-7]])
>>> print(solution_3values)
[-0.8611, -1.3889, -0.0278]