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 __________.

Consider matrix $$A = \left[ {\matrix{ k & {2k} \cr {{k^2} - k} & {{k^2}} \cr } } \right]$$ and

vector $$X = \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$.

The number of distinct real values of k for which the equation AX = 0 has infinitely many solutions is _______.

Let M be a real 4 $$ \times $$ 4 matrix. Consider the following statements :

S1: M has 4 linearly independent eigenvectors.

S2: M has 4 distinct eigenvalues.

S3: M is non-singular (invertible).

Which one among the following is TRUE?

The rank of the matrix $$M = \left[ {\matrix{ 5 & {10} & {10} \cr 1 & 0 & 2 \cr 3 & 6 & 6 \cr } } \right]$$ is