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&amp;1&amp;0\\1&amp;c&amp;1\\0&amp;1&amp;c\end{pmatrix}
$R_1,R_2,R_3$ are the first, second and third row

Let's assume that $c\neq0$
, as the case $c=0$ is very simple.

$\begin{pmatrix}c&amp;1&amp;0\\1&amp;c&amp;1\\0&amp;1&amp;c\end{pmatrix}$ $R_1\leftarrow \frac{R_1}c$

$\begin{pmatrix}1&amp;1/c&amp;0\\1&amp;c&amp;1\\0&amp;1&amp;c\end{pmatrix}$ $R_2\leftarrow R_2-R_1$

$\begin{pmatrix}1&amp;1/c&amp;0\\0&amp;c-1/c&amp;1\\0&amp;1&amp;c\end{pmatrix}$ $R_2\leftarrow R_2\cdot\frac{c}{c^2-1}$

$\begin{pmatrix}1&amp;1/c&amp;0\\0&amp;1&amp;\frac{c}{c^2-1}\\0&amp;1&amp;c\end{pmatrix}$ $R_3\leftarrow R_3-R_2$

$\begin{pmatrix}1&amp;1/c&amp;0\\0&amp;1&amp;\frac{c}{c^2-1}\\0&amp;0&amp;c-\frac{c}{c^2-1}\end{pmatrix}$ $R_1\leftarrow R_1-c\cdot R_2$

$\begin{pmatrix}1&amp;0&amp;\frac{c^2}{c^2-1}\\0&amp;1&amp;\frac{c}{c^2-1}\\0&amp;0&amp;\frac{c^3-2c}{c^2-1}\end{pmatrix}$

Can you do the last two steps by yourself ? :) :) :)
Find the determinant of this matrix and notice where it becomes zero. The matrix is invertible for every value except those.