TITLE: subspaces of singular matrices
QUESTION [12 upvotes]: Let $A$, $B$ be square matrices over infinite field (we identify them with linear operators on the vector space of columns). It is given that for all scalars $a,b$ the matrix $aA+bB$ is singular. Does it follow that there exist matrices $P$, $Q$ such that rank$(P)$+rank$(Q) > n$ but $PAQ=PBQ=0$?
If yes, is the same true for arbitrary subspaces of singular matrices? Well, the answer is no for antisymmetric matrices $3\times 3$... But how can subspaces of singular matrices be described (if they can)?

REPLY [2 votes]: `But how can subspaces of singular matrices be described (if they can)?'

I doubt they can. For instance, there is the following counter-intuitive result:

for infinitely many $n$, there exists
an 8-dimensional space of $n\times n$ matrices
(over any field of characteristic zero) that is maximal singular (i.e. inclusion-maximal subspace of singular matrices).