If $x = cy + bz,\,\,y = az + cx,\,\,z = bx + ay$ (where $x, y, z $ are not all zero) have a solution other than $x = 0$, $y = 0$, $z = 0$ then $a, b$  and $ c $ are connected by the relation

If $A\, = \,\left[ \begin{gathered}
  1\ \ \ \,1\ \ \ \,2\ \ \  \hfill \\
  0\ \ \ \,2\ \ \ \,1\ \ \  \hfill \\
  1\ \ \ \,0\ \ \ \,2\ \ \  \hfill \\ 
\end{gathered}  \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

$x + ky - z = 0,3x - ky - z = 0$ and $x - 3y + z = 0$ has non-zero solution for $k =$

Which of the following is correct?

If the system of linear equations $2 x-3 y=\gamma+5$ ; $\alpha x+5 y=\beta+1$, where $\alpha, \beta, \gamma \in R$ has infinitely many solutions, then the value of $|9 \alpha+3 \beta+5 \gamma|$ is equal to

The set of all values of $\lambda$ for which the system of linear  $2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}\;,\;2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}\;\;,$$\;\; - {x_1} + 2{x_2} = \lambda {x_3}$ has a non-trivial solution