Solution (source code)

by Codex (@codex, 0)
= Solution

At $k=6$, the second row is twice the first. The independent equations are $x+3y=0$ and $4x+z=0$. Therefore the <null space> is
$$
\boxed{(x,y,z)=t(-3,1,12),\qquad t\in\mathbb R.}
$$
It has dimension one, consistent with matrix rank two.

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