Is it true that two $3 \times 3$ matrices in $R$ are similar $\iff$ they have same determinant and same trace? Is it easy to cite some example of $3 \times 3 $ non-similar matrices of real entries with equal determinant and trace?

Even simpler examples than Chappers' one would be
$$ \begin{pmatrix} 0&0&0\\0&0&0\\0&0&0 \end{pmatrix} \quad\text{and}\quad
\begin{pmatrix} 0&0&1\\0&0&0\\0&0&0 \end{pmatrix} $$
(determinant and trace both $0$, but the zero matrix is similar only to itself) or
$$ \begin{pmatrix} 1&0&0\\0&1&0\\0&0&1 \end{pmatrix} \quad\text{and}\quad
\begin{pmatrix} 1&1&0\\0&1&1\\0&0&1 \end{pmatrix} $$
(determinant $1$, trace $3$, but the identity matrix is similar only to itself).
