Editing System of linear equations (section)

===Matrix solution===
If the equation system is expressed in the matrix form <math>A\mathbf{x}=\mathbf{b}</math>, the entire solution set can also be expressed in matrix form. If the matrix ''A'' is square (has ''m'' rows and ''n''=''m'' columns) and has full rank (all ''m'' rows are independent), then the system has a unique solution given by

: <math>\mathbf{x}=A^{-1}\mathbf{b}</math>

where <math>A^{-1}</math> is the [[matrix inverse|inverse]] of ''A''. More generally, regardless of whether ''m''=''n'' or not and regardless of the rank of ''A'', all solutions (if any exist) are given using the [[Moore–Penrose inverse]] of ''A'', denoted <math>A^+</math>, as follows:

: <math>\mathbf{x}=A^+ \mathbf{b} + \left(I - A^+ A\right)\mathbf{w}</math>

where <math>\mathbf{w}</math> is a vector of free parameters that ranges over all possible ''n''×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using <math>\mathbf{w}=\mathbf{0}</math> satisfy <math>A\mathbf{x}=\mathbf{b}</math> &mdash; that is, that <math>AA^+ \mathbf{b}=\mathbf{b}.</math> If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which ''A'' is square and of full rank, <math>A^+</math> simply equals <math>A^{-1}</math> and the general solution equation simplifies to
: <math>\mathbf{x}=A^{-1}\mathbf{b} + \left(I - A^{-1}A\right)\mathbf{w} = A^{-1}\mathbf{b} + \left(I-I\right)\mathbf{w} = A^{-1}\mathbf{b}</math>
as previously stated, where <math>\mathbf{w}</math> has completely dropped out of the solution, leaving only a single solution. In other cases, though, <math>\mathbf{w}</math> remains and hence an infinitude of potential values of the free parameter vector <math>\mathbf{w}</math> give an infinitude of solutions of the equation.