<s>[INST] Sum of matrices value Let $\mspace{10mu}A,B\in M^{4\times 4}(\mathbb{R})\mspace{10mu}, $
$A=\begin{pmatrix}
1 &0  &2  &-1 \\ 
0 &1  &-1  &1 \\ 
1 &1  &1  &0 \\ 
1 &-1  &3  &-2 
\end{pmatrix} \mspace{10mu} \mathrm{and} \mspace{10mu} \mathrm{rk}(B)=1$ 
What values may have   $\mspace{10mu}\mathrm{rk}(A+B)\mspace{10mu}?$
I only figured out that $\mathrm{rk}(A)=2$.
 [/INST] It cannot be $A = -B$ since $\operatorname{rank}(A) = 2$ and $\operatorname{rank}(B) = 1$ so $A+B \ne 0$. Therefore by the subadditivity of rank we have
$$1 \le \operatorname{rank}(A+B) \le \operatorname{rank}(A) + \operatorname{rank}(B) = 3$$
so $\operatorname{rank}(A+B) \in \{1,2,3\}$.
All possibilities are indeed attainable, consider $B$ equal to
$$\begin{pmatrix}
0 &0  &0  &0 \\ 
0 &0  &0  &0 \\ 
0 &0  &1  &0 \\ 
0 &0  &0  &0 
\end{pmatrix}, \quad\begin{pmatrix}
0 &0  &0  &0 \\ 
1 &1  &1  &1 \\ 
1 &1  &1  &1 \\ 
-1 &-1  &-1  &-1 
\end{pmatrix}, \quad\begin{pmatrix}
0 &0  &0  &0 \\ 
0 &-1  &1  &-1 \\ 
0 &-1  &1  &-1 \\ 
0 &1  &-1  &1 
\end{pmatrix}$$
respectively.
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