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Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
The Gaussian method is a method to solve a given system of linear equations (SLE) written in the form of an extended coefficient matrix:
$$A\beta:=\left(\begin{array}{cccc}\alpha_{11}& \ldots&\alpha_{1n}&\beta_1\\
\alpha_{21}& \ldots&\alpha_{2n}&\beta_2\\
\vdots&\vdots&\vdots&\vdots\\
\alpha_{m1}& \ldots&\alpha_{mn}&\beta_m\end{array}\right)\quad\quad ( * )$$
with $\beta_i,a_ij\in F$, where $F$ is a field and not all $\beta_i,a_{ij}$ are zero^{1}.
The method consists of the following two steps:
 Bring the system $( * )$ to an upper triangular form
$$\left(\begin{array}{cccccccc}a_{11}& a_{12}&\ldots&a_{1r}&a_{1,r+1}&\ldots&a_{1n}&b_1\\
0& a_{22}&\ldots&a_{2r}&a_{2,r+1}&\ldots&a_{2n}&b_2\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\
0& 0 &\ldots&a_{rr}&a_{r,r+1}&\ldots&a_{rn}&b_r\\
\vdots& \vdots &\ldots&0&0&\ldots&0&b_{r+1}\\
\vdots& \vdots &\ldots&\vdots&\vdots&\ldots&\vdots&\vdots\\
0& \ldots &\ldots&0&0&\ldots&0&b_m\\
\end{array}\right)\quad \quad ( * * )$$
using the elementary Gaussian operations, where^{2} all the $a_{ii}\neq 0$ for $i=1,\ldots$ with some $r\in\{1,\ldots,\min(m,n)\}.$ The number $r$ is called the rank of the coefficient matrix $A.$
The first step might involve the following steps:
 Remove all columns consisting of only zero elements.
 Set $k=1$.
 Check, if in the remaining system, the element $\alpha_{kk}$ is nonzero, if so proceed with the step 5.
 Let $i$ be the row, in which in the first column the element $\alpha_{ik}$ is not zero. Exchange the rows $1$ and $i,$ i.e. set $a_{1k}:=\alpha_{ik}$ (applying Gaussian operation $I.$)^{3}
 To set the numbers $\alpha_{2k},\ldots,\alpha_{mk}$ to zero, add to the rows $2,\ldots,m$ a suitable multiple of the $k$th row (applying Gaussian operation $III.$)
 If the upper triangular form $( * * )$ has been reached, end.
 Otherwise set $k=k+1$ and go back to step 3.
 If in the system $( * * )$ $b_j\neq 0$ for at least one $j > r,$ then the system $( * )$ has no solutions^{4}. Otherwise, we can solve the system $( * * )$ using the backward substitution.
The Gaussian method makes use of the lemma that the elementary Gaussian operations do not change the set of solutions of the system $( * ).$ Thus, the solution to $( * * )$ found in the second step is also a solution to $( * ).$
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Examples: 1
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References
Bibliography
 Knabner, P; Barth, W.: "Lineare Algebra  Grundlagen und Anwendungen", Springer Spektrum, 2013
Footnotes