Matrix of Cofactors

matrix_of_cofactors(matrix)

Create the matrix of cofactors corresponding to a given matrix

Parameters

matrix (list of lists of int or float) – List of lists of numbers representing a matrix

Raises

  • TypeError – Argument must be a 2-dimensional list

  • TypeError – Elements nested within argument must be integers or floats

Returns

matrix – List of lists in which each inner element alternates being positive or negative versions of the corresponding element from the original matrix

Return type

list of lists of int or float

See also

matrix_of_minors(), transposed_matrix(), linear_determinant(), inverse_matrix()

Notes

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

  • Matrix of cofactors (if \(\mathbf{A}\) contains an odd number of rows and columns): \(\mathbf{A}^C = \begin{bmatrix} a_{1,1} & -1\cdot{a_{1,2}} & \cdots & a_{1,n} \\ -1\cdot{a_{2,1}} & a_{2,2} & \cdots & -1\cdot{a_{2,n}} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m,1} & -1\cdot{a_{m,2}} & \cdots & a_{m,n} \end{bmatrix}\)

  • Matrix of cofactors (if \(\mathbf{A}\) contains an even number of rows and columns): \(\mathbf{A}^C = \begin{bmatrix} a_{1,1} & -1\cdot{a_{1,2}} & \cdots & -1\cdot{a_{1,n}} \\ -1\cdot{a_{2,1}} & a_{2,2} & \cdots & a_{2,n} \\ \cdots & \cdots & \cdots & \cdots \\ -1\cdot{a_{m,1}} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}\)

  • Matrix of Cofactors

Examples

Import matrix_of_cofactors function from regressions library
>>> from regressions.matrices.cofactors import matrix_of_cofactors
Create the matrix of cofactors for [[1, 2, 3], [4, 5, 6]]
>>> matrix_3x2 = matrix_of_cofactors([[1, 2, 3], [4, 5, 6]])
>>> print(matrix_3x2)
[[1, -2, 3], [-4, 5, -6]]
Create the matrix of cofactors for [[2, 3], [5, 7]]
>>> matrix_2x2 = matrix_of_cofactors([[2, 3], [5, 7]])
>>> print(matrix_2x2)
[[2, -3], [-5, 7]]