  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
$n=2$, 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.
