Consider the matrix $A$ below:

$$ A=\left[\begin{array}{llll} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{array}\right] $$

For which of the following combinations of $\alpha, \beta$ and $\gamma$, is the rank of $A$ at least three?

(i) $\alpha=0$ and $\beta=\gamma \neq 0$

(ii) $\alpha=\beta=\gamma=0$

(iii) $\beta=\gamma=0$ and $\alpha \neq 0$

(iv) $\alpha=\beta=\gamma \neq 0$

Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors

$$ \{ [2 \ -3 \ \alpha], \ [3 \ -1 \ 3], \ [1 \ -5 \ 7] \}$$

does not form a basis of $\mathbb{R}^3$ is _______.

Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are

Consider a system of linear equations Ax = b, where

$$A = \left[ {\matrix{ 1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr { - 1} \hfill & {\sqrt 2 } \hfill & { - 3} \hfill \cr } } \right]$$, $$b = \left[ {\matrix{ 1 \cr 3 \cr } } \right]$$

This system is equations admits __________.