Find the value(s) of $c$ such that the following linear system is inconsistent \begin{align} x_1 + x_2 + cx_3 &= 0 \\ x_2 + x_3 &= 0 \\ cx_1 + x_3 &= 1 \end{align}
I just started learning about linear systems and I'm really confused on how to start on this problem. I'm trying to make a row to be  $\begin{pmatrix} 0 & 0 & 0 &=& 1 \end{pmatrix}$ so that its inconsistent but I'm not sure where to start.
Thanks a lot!!
 A: For a inconsistent system,
$$ \begin{vmatrix} 
1 & 1 & c \\
0 & 1 & 1 \\
c & 0 & 1  
\end{vmatrix} = 0
$$
and atleast one of $ \begin{vmatrix} 
0 & 1 & c \\
0 & 1 & 1 \\
1 & 0 & 1  
\end{vmatrix}$, $ \begin{vmatrix} 
1 & 0 & c \\
0 & 0 & 1 \\
c & 1 & 1  
\end{vmatrix}$ or $ \begin{vmatrix} 
1 & 1 & 0 \\
0 & 1 & 0 \\
c & 0 & 1  
\end{vmatrix}$ must be non-zero.
A: Let's compute a row echelon form of the system’s matrix
\begin{align}
\begin{bmatrix}
1 & 1 & c & 0 \\
0 & 1 & 1 & 0 \\
c & 0 & 1 & 1
\end{bmatrix}
&\to
\begin{bmatrix}
1 & 1 & c & 0 \\
0 & 1 & 1 & 0 \\
0 & -c & 1-c^2 & 1
\end{bmatrix}
&& R_3\gets R_3-cR_1
\\[6px]&\to
\begin{bmatrix}
1 & 1 & c & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1+c-c^2 & 1
\end{bmatrix}
&& R_3\gets R_3+cR_2
\end{align}
The system is inconsistent if and only if $1+c-c^2=0$.

You might also try and solve the system by noticing that $x_2=-x_3$ and $x_3=1-cx_1$, so the first equation becomes
$$
x_1-(1-cx_1)+c(1-cx_1)=0
$$
or
$$
(1+c-c^2)x_1=1-c
$$
A: Let's row reduce the augmented matrix associated with the system: $\left(\begin{array}{rrr|r}1&1&c&0\\0&1&1&0\\c&0&1&1\end{array}\right)\to\left(\begin{array}{rrr|r}1&1&c&0\\0&1&1&0\\0&-c&1-c^2&1\end{array}\right)\to\left(\begin{array}{rrr|r}1&1&c&0\\0&1&1&0\\0&0&1+c-c^2&1\end{array}\right)$.
So to answer the question,  $1+c-c^2=0$ gives an inconsistent system. So $c=\frac{1\pm\sqrt5}2$.
