Solution
= Solution
Let
$$
U=(e_3,e_1,e_2)
=
\begin{pmatrix}
0&1&0\\
0&0&1\\
1&0&0
\end{pmatrix}.
$$
This permutation <matrix> is unitary. The actions
$$
Ae_3=e_3,\qquad
Ae_1=2e_3+4e_1,\qquad
Ae_2=3e_3+5e_1+6e_2
$$
give
$$
\boxed{
U^\dagger AU=
\begin{pmatrix}
1&2&3\\
0&4&5\\
0&0&6
\end{pmatrix}},
$$
which is upper triangular.