In the proceeding examples, in particular, the example of backward substitution, we have seen that if an SLE is in an upper triangular form,
$$\left(\begin{array}{ccccccc|c}\alpha_{11}& \alpha_{12}&\ldots&\alpha_{1r}&\alpha_{1,r+1}&\ldots&\alpha_{1n}&\beta_1\\ 0& \alpha_{22}&\ldots&\alpha_{2r}&\alpha_{2,r+1}&\ldots&\alpha_{2n}&\beta_2\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0& 0 &\ldots&\alpha_{rr}&\alpha_{r,r+1}&\ldots&\alpha_{rn}&\beta_r\\ \vdots& \vdots &\ldots&0&0&\ldots&0&\beta_{r+1}\\ \vdots& \vdots &\ldots&\vdots&\vdots&\ldots&\vdots&\vdots\\ 0& \ldots &\ldots&0&0&\ldots&0&\beta_m\\ \end{array}\right)$$
then we can easily decide whether there is no solution to the system or we can find an explicit solution to it. Therefore, if we could find some way to transform any given SLE into an upper triangular form, without changing the set of its solutions, then this would enable us to solve any given SLE, provided that it has a solution. Indeed, such a transformation exists and was found by the German mathematician Carl Friedrich Gauss (1777 - 1855). In the following, we define elementary Gaussian operations allowing us to transform any given SLE into an upper triangular form.
Examples: 1