Let $A\in\mathbb{M}_n(K)$. What are the possible values of $$\text{rank}(A+\text{adj}(A))?$$ (Since this question was not in here and its answer needs some thinking, I'm adding it in a Q&A style)

Resolved The tag is now adjoint-operators; a tag adjoint-functors was also created The tag adjoint is "[f]or questions about adjoints, in the category-theoretic or inner-product-space sense, as well as about adjugate matrices", which certainly falls under "two or more completely unrelated th...