Try using the Gauss Jordan Method to get the matrix in Reduced Row Echelon Form:
Start with $i=1$, $j=1$
If $a_{ij}=0$ swap the $i$-th row with some other row to guarantee that $a_{ij}\neq 0$. If all the entries in the column are zero, increase $j$ by $1$
Divide the $i$-th row by $a_{ij}$ to make the pivot entry $1$
Eliminate all other entries in the $j$-th colum by subtracting suitable multiples of the $i$-th row from the other rows
Increase $i$ by $1$ and $j$ by $1$. Return to Step $1$
The algorithm stops after we process the last row or the last column of the matrix
Can you continue from here?
Edit: We can use an online calculator to see that the RREF has no one row of zeros so I would tentatively say that there is no value of $k$ which gives a rank of $2$ (I may be wrong here though, it's been a while since I did this stuff). Have you definitely written down/typed the matrix correctly?