<s>[INST] How to build a linear equation system? How can one build a $3\times 3$ linear equation system ($3$ equations with $3$ variables) if the variables are known, for example, $a=1, b=2$ and $c=3$?
Thank you in advance!
 [/INST] Another possibility is to start with a non-singular upper-left triangular matrix $T$ and the desired solution $\vec{x}=(x_1,x_2,x_3)^t$. Now one adds multiples of upper rows to lower ones to obtain a full matrix $A$, e.q.
\begin{gather*}
  T=\begin{pmatrix}1& 2 & 3\cr 0& 4 & 5\cr 0 & 0 & 6\end{pmatrix},\; 
    \vec{x}=\begin{pmatrix}1\cr 2\cr3\end{pmatrix}\\
  T_1=\mbox{3rd row}+2\cdot\mbox{2nd row}
     =\begin{pmatrix}1& 2 & 3\cr 0& 4 & 5\cr 0 & 8 & 16\end{pmatrix}\\
  T_2=\mbox{2nd row}+3\cdot\mbox{1st row}
     =\cdots\\
  T_3=\mbox{3nd row}+5\cdot\mbox{1st row}
     =\begin{pmatrix}1& 2 & 3\cr 3& 10 & 14\cr 5 & 18 & 31\end{pmatrix}
\end{gather*}
If the desired solution ist $\vec{x}=(1,2,3)^t$ then $T\cdot \vec{x}$ yields the right hand side $\vec{b}=(14,\,23,\,18)^t$. Transforming $\vec{b}$ according to $T$ gives the linear system
\begin{gather*}
\begin{pmatrix}1& 2 & 3\cr 3& 10 & 14\cr 5 & 18 & 31\end{pmatrix}
\begin{pmatrix}x\cr y\cr z\end{pmatrix}
=\begin{pmatrix}14\cr 65\cr 134\end{pmatrix}
\end{gather*}
Advantage of this method is that there exists a solution for the students with 'easy' numbers.
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