The set of all values of $t\in \mathrm{ℝ}$, for which the matrix $\left[\begin{array}{ccc}{\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)& {\mathrm{e}}^{-t}\left(-2\sin t-\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(2\sin t+\cos t\right)& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\cos t& {\mathrm{e}}^{-t}\sin t\end{array}\right]$ is invertible, is

Let the system of linear equations

$x+y+kz=2$

$2x+3y-z=1$

$3x+4y+2z=k$

have infinitely many solutions. Then the system

$\left(k+1\right) x+\left(2k-1\right) y=7$

$\left(2k+1\right)x+\left|k+5\right|y=10$ has :

For $\alpha ,\beta \in \mathrm{ℝ}$, suppose the system of linear equations
$x-y+z=5$ 
$2x+2y+\alpha z=8$
$3x-y+4z=\beta$

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of 

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations

$\lambda x+y+z=1$

$x+\lambda y+z=1$

$x+y+\lambda z=1$
is inconsistent, then $\sum _{\lambda \in S}\left({\left|\lambda \right|}^2+\left|\lambda \right|\right)$ is equal to