$\text{rank}(A) = 3$ and $\text{nullity}(A) = 2$

$A\mathbf{x} = \mathbf{b}$ has a unique solution for all $\mathbf{b}$

$A\mathbf{x} = \mathbf{0}$ has only the trivial solution $\mathbf{x} = \mathbf{0}$

$\text{rank}(A) = 3$ and there are 2 non-pivot columns