Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a realvalued constant. Which of the following option(s) is/are CORRECT?

$$\begin{aligned} & x+k y=1 \\ & k x+y=-1 \end{aligned}$$

The product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is

The Lucas sequence $$L_n$$ is defined by the recurrence relation:

$${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$,

with $${L_1} = 1$$ and $${L_2} = 3$$.

Which one of the options given is TRUE?

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \right]$$.

Let $$\mathrm{det}(A)$$ and $$\mathrm{det}(B)$$ denote the determinates of the matrices A and B, respectively.

Which one of the options given below is TRUE?