Editing Cramer's rule (section)

==General case==
Consider a system of {{mvar|n}} linear equations for {{mvar|n}} unknowns, represented in matrix multiplication form as follows:

:<math> A\mathbf{x} = \mathbf{b}</math>

where the {{math|''n'' × ''n''}} matrix {{mvar|A}} has a nonzero determinant, and the vector <math> \mathbf{x} = (x_1, \ldots, x_n)^\mathsf{T} </math> is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:

:<math> x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, \ldots, n</math>

where <math> A_i </math> is the matrix formed by replacing the {{mvar|i}}-th column of {{mvar|A}} by the column vector {{math|'''b'''}}.

A more general version of Cramer's rule<ref>{{cite journal |author1=Zhiming Gong |author2=M. Aldeen |author3=L. Elsner |title=A note on a generalized Cramer's rule |journal=Linear Algebra and Its Applications |volume=340 |year=2002 |issue=1–3 |pages=253–254 |doi=10.1016/S0024-3795(01)00469-4|doi-access=free }}</ref> considers the matrix equation

:<math> AX = B</math>

where the {{math|''n'' × ''n''}} matrix {{mvar|A}} has a nonzero determinant, and {{mvar|X}}, {{mvar|B}} are {{math|''n'' × ''m''}} matrices. Given sequences <math> 1 \leq i_1 < i_2 < \cdots < i_k \leq n </math> and <math> 1 \leq j_1 < j_2 < \cdots < j_k \leq m </math>, let <math> X_{I,J} </math> be the {{math|''k'' × ''k''}} submatrix of {{mvar|X}} with rows in <math> I := (i_1, \ldots, i_k ) </math> and columns in <math> J := (j_1, \ldots, j_k ) </math>. Let <math> A_{B}(I,J) </math> be the {{math|''n'' × ''n''}} matrix formed by replacing the <math>i_s</math> column of {{mvar|A}} by the <math>j_s</math> column of {{Mvar|B}}, for all <math> s = 1,\ldots, k </math>. Then

:<math> \det X_{I,J} = \frac{\det(A_{B}(I,J))}{\det(A)}. </math>

In the case <math> k = 1 </math>, this reduces to the normal Cramer's rule.

The rule holds for systems of equations with coefficients and unknowns in any [[field (mathematics)|field]], not just in the [[real number]]s.