Matrices with prescribed nonprincipal blocks and the trace function (Q757527)

Let A and B be two \(n\times n\) matrices over an arbitrary field. The author shows that for \(n\geq 3\), there exist invertible matrices U and V such that \(VAU^{-1}+UBV^{-1}\) is nonscalar and has a prescribed trace. When \(n=2\), the result fails only when the field has two or three elements and A and B are certain matrices.