Simple question on vector subspace complements and matrix rank

Question

Is it the following statement correct?

Given three matrices $A, B, C \in \Bbb R^{N\times N}$, if $\def\rank{\operatorname{rank}}\rank(A)=N>\rank(B)$ (so that the columns of $A$ span the whole space $\Bbb R^{N}$ while those of $B$ just span a subspace of $\Bbb R^{N}$), and $\rank(A)=N=\rank(B+C)$, then the columns of $C$ span the orthogonal complement of the subspace spanned by those of $B$.

Edit: what if, in addition to $\rank(A)=N=\rank(B+C)$, we also impose A=B+C? Is it true in this case that the columns of $C$ span the orthogonal complement of the subspace spanned by those of $B$?

Thanks

Answer 1

It is false. Take, for instance, $N=2$, $A=\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]$, $B=\left[\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right]$, and $C=\left[\begin{smallmatrix}0&1\\0&1\end{smallmatrix}\right]$.