A is a 3 × 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?

If the sum and product of eigenvalues of a 2 × 2 real matrix $\begin{bmatrix}3&p\\\ p&q\end{bmatrix} $ are 4 and -1 respectively, then |p| is _______ (in integer).

The system of linear equations in real (x, y) given by

$\rm \begin{pmatrix} \rm x & \rm y \end{pmatrix} \begin{bmatrix} 2 & 5- 2 α \\\ α & 1 \end{bmatrix} = \rm \begin{pmatrix} \rm 0 & \rm 0 \end{pmatrix} $

involves a real parameter α and has infinitely many non-trivial solutions for special value(s) of α. Which one or more among the following options is/are non-trivial solution(s) of (x, y) for such special value(s) of α ?

Consider the matrix $$A = \left[ {\matrix{ {50} & {70} \cr {70} & {80} \cr } } \right]$$ whose eigenvectors corresponding to eigen values $${\lambda _1}$$ and $${\lambda _2}$$ are $${x_1} = \left[ {\matrix{ {70} \cr {{\lambda _1} - 50} \cr } } \right]$$ and $${x_2} = \left[ {\matrix{ {{\lambda _2} - 80} \cr {70} \cr } } \right],$$ respectively. The value of $$x_1^T{x_2}$$ is ________