<s>[INST] Check the invertibility of a matrix given a parameter Find all values of $c$, if any, for which the given matrix is invertible (using row operations -> gotta try simplifying to I). 
Here is the matrix : 
\begin{pmatrix}c&1&0\\1&c&1\\0&1&c\end{pmatrix}
 [/INST] $R_1,R_2,R_3$ are the first, second and third row
Let's assume that $c\neq0$
, as the case $c=0$ is very simple.
$\begin{pmatrix}c&1&0\\1&c&1\\0&1&c\end{pmatrix}$ $R_1\leftarrow \frac{R_1}c$
$\begin{pmatrix}1&1/c&0\\1&c&1\\0&1&c\end{pmatrix}$ $R_2\leftarrow R_2-R_1$
$\begin{pmatrix}1&1/c&0\\0&c-1/c&1\\0&1&c\end{pmatrix}$ $R_2\leftarrow R_2\cdot\frac{c}{c^2-1}$
$\begin{pmatrix}1&1/c&0\\0&1&\frac{c}{c^2-1}\\0&1&c\end{pmatrix}$ $R_3\leftarrow R_3-R_2$
$\begin{pmatrix}1&1/c&0\\0&1&\frac{c}{c^2-1}\\0&0&c-\frac{c}{c^2-1}\end{pmatrix}$ $R_1\leftarrow R_1-c\cdot R_2$
$\begin{pmatrix}1&0&\frac{c^2}{c^2-1}\\0&1&\frac{c}{c^2-1}\\0&0&\frac{c^3-2c}{c^2-1}\end{pmatrix}$
Can you do the last two steps by yourself ? :) :) :)
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