Given two non-commuting matrices $A,B$, can we always find another nontrivial matrix $C$ that commute with each of them and how to construct $C$ explicitly?
This is sort of the converse statement of that fact that commutation relation are not transitive. If the statement is not generally true, what minimal constraints could be added such that it is true?
For the matrices $$A=\pmatrix{1&1\\0&1}, B=\pmatrix{1&0\\1&1}$$ it's easy to check that only the scalar matrices commute with both $A$ and $B$. Indeed those that commute with $A$ are of the form $$\pmatrix{a&b\\0&a},$$ and those that commute with $B$ are of the form $$\pmatrix{a&0\\c&a}$$