2x2-matrix/Trace is 0/Anti diagonal/Exercise

Let

${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,$

be a matrix over ${}\mathbb {Q}$, whose trace is ${}0$. Show that there exists a matrix ${}N$ of the form

${}N={\begin{pmatrix}0&r\\s&0\end{pmatrix}}\,$

that is similar to ${}M$.

Create a solution