Question:
Suppose that the linear system $$\begin{bmatrix} 2 & 0 & -1 \\ 0 & c & 1 \\ 1 & 3 & -2 \end{bmatrix} \vec{x}=\vec{b}= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ is inconsistent for some vector $\vec{b}$. Determine c.
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What I did:
I put this into RREF and got :
$$\left[\begin{array}{rrr|r} 2 & 0 & -1 & x_1\\ 0 & c & 1 & x_2 \\ 0 & 2+c & 0 & 2x_3-x_1+x_2 \end{array}\right]$$
I do not know what to do next.
The system is inconsistent for some vector $b$ if and only if its rank is $<3$, since any system of rank $3$ is consistent, i. e. the linear map associated with the matrix of the system is an isomorphism.
Let's row reduce the matrix of the system: $$\begin{bmatrix}2&0&-1\\0&c&1\\1&3&-2\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&c&1\\0&6&-3\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&6&-3\\0&c&1\end{bmatrix}\rightsquigarrow\begin{bmatrix}2&0&-1\\0&6&-3\\0&0&3c+6\end{bmatrix} $$ We see the system has rank $<3$ (and indeed has rank $2$) if and only if $3c+6=0$, i. e. $\,c=-2$.
