\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!!
Best Answer
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 $$ (1+c-c^2)x_1=1-c $$