Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v₁ and v₂ satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :

A = $$\alpha$$v₁$$v_1^T$$ + $$\beta$$v₂$$v_2^T$$

The eigen values of A are __________.

A real 2 $$\times$$ 2 non-singular matrix A with repeated eigen value is given as

$$A = \left[ {\matrix{ x & { - 3.0} \cr {3.0} & {4.0} \cr } } \right]$$

where x is a real positive number. The value of x (rounded off to one decimal) is _______

Consider the following system of linear equations.

$$ x_1+2 x_2=b_1 ; 2 x_1+4 x_2=b_2 ; 3 x_1+7 x_2=b_3 ; 3 x_1+9 x_2=b_4 $$

Which one of the following conditions ensures that a solution exists for the above system?

The rank of the matrix $$\left[ {\matrix{ 1 & { - 1} & 0 & 0 & 0 \cr 0 & 0 & 1 & { - 1} & 0 \cr 0 & 1 & { - 1} & 0 & 0 \cr { - 1} & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 1 & { - 1} \cr } } \right]$$ is __________.