Gaussian elimination method
Determine the value of $A$ which makes the system incompatible $$\left\{ \begin{array} {rcl} x+Ay+z & = & 1 \\ Ax+y+(A-1)z &=& A\\ x+y+z &=& A+1\end{array}\right.$$
The first thing is to re-write the system in matricial form: $$\begin{pmatrix} 1 & A & 1 & | & 1 \\ A & 1 & (A-1) & | & A \\ 1 & 1 & 1 & | & A+1 \end{pmatrix}$$ We then use the Gauss method $$\begin{pmatrix} 1 & A & 1 & | & 1 \\ A & 1 & (A-1) & | & A \\ 1 & 1 & 1 & | & A+1 \end{pmatrix} \rightarrow \left\{ \begin{array}{c} r2-Ar1 \\ r3-r1 \end{array} \right. \rightarrow \begin{pmatrix} 1 & A & 1 & | & 1 \\ 0 & 1-A^2 & -1 & | & 0 \\ 0 & 1-A & 0 & | & A \end{pmatrix} \rightarrow $$ $$\rightarrow (c3 \leftrightarrow c2) \rightarrow \begin{pmatrix} 1 & 1 & A & | & 1 \\ 0 & -1 & 1-A^2 & | & 0 \\ 0 & 0 & 1-A & | & A \end{pmatrix}$$ We obtained: $(1-A)y=A$ *, then:
*See that, as we changed the columns, we obtain the solution for $y$ and not for $z$.
If $A=1$, we have an incompatible system.
Otherwise the system is compatible, and it has the following solution:
$$y=\dfrac{A}{1-A}; \ z=(1+A)A; \ x=\dfrac{A^3-A^2-2A+1}{1-A}$$
$$A=1$$