Kernel of a matrix

The second row is twice the first, so the rows span a line and $\operatorname{rank}A=1$.

The kernel is the set of $(x,y,z)\in\mathbb{R}^3$ with $x+2y+3z=0$ (the second equation is a multiple of the first). Writing $x=-2y-3z$,
$$(x,y,z)=y\,(-2,1,0)+z\,(-3,0,1).$$

So $\{(-2,1,0),(-3,0,1)\}$ is a basis of $\ker A$, which has dimension $2=3-\operatorname{rank}A$, as the rank–nullity theorem predicts.