We describe transposed Poisson structures on the upper triangular matrix Lie algebra T n .F /, n > 1, over a field F of characteristic zero. We prove that, for n > 2, any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n D 2, there is one more class of transposed Poisson structures on T n .F /. We also show that, up to isomorphism, the full matrix Lie algebra M n .F / admits only one non-trivial transposed Poisson structure, and it is of Poisson type.