An example of matrices $A_1$ and $A_2$, for which the rank of sum $A_1+A_2$ is greater than $max(r_1, r_2)$

Question

Let's assume that the matrices $A_1$, $A2$ $M_{n×n}(\mathbb{Q})$ are of rank $r_1$ and $r_2$ respectively. Give an example of matrices $A_1$ and $A_2$, for which the rank of sum $A_1+A_2$ is greater than $max(r_1, r_2)$. Give an example of $A_1$ and $A_2$ for which the rank of sum $A_1+A_2$ is smaller than $min(r_1, r_2)$.

Is that possible? I tried to do the first part and I allways get the rank equal to $max(r_1, r_2)$. So I am not sure if there is an answer to that question.

Answer 1

Hint: Consider $A_1+A_2=I$. For the second one, consider subtraction rather than addition.