(related to Section: Solving Simple Systems of Linear Equations)
The next example is a more interesting SLE. While it is quite simple, it has non-trivial solutions. Its extended coefficient matrix contains a diagonal matrix. $$\left(\begin{array}{cccc|c}\alpha_{11}& 0&\ldots&0&\beta_1\\ 0& \alpha_{22}&\ldots&0&\beta_2\\ \vdots&\vdots&\ddots&\vdots\\ 0& 0 &\ldots&\alpha_{mm}&\beta_m\end{array}\right)$$
in which we assume all $\alpha_jj\neq 0$ for $j=1,\ldots,m.$ Please note that the coefficient matrix here is a square matrix, since it has $m$ columns and $m$ rows. Each linear equation in such an SLE has the form
$$\begin{array}{rcl}0x_1+\ldots+0x_{j-1}+\alpha_{jj}x_j+0x_{j+1}+\ldots+0x_n&=&\beta_j\\ \alpha_{jj}x_j&=&\beta_j\end{array}$$ and therefore the SLE has the solution $$x_j:=\frac{\beta_j}{\alpha_{jj}}$$ for $j=1,\ldots,m.$
This example shows us that a diagonal SLE is not really a system of linear equations since the simultaneous solutions to all equations are independent of each other. In other words, the system does not pose any constraints to the solution of every single equation which would depend on other equations.
Examples: 1