Matrix product/Upper triangular matrix/5/Zero diagonal/Nilpotent/Exercise

We consider the matrix

${}M={\begin{pmatrix}0&a&b&c&d\\0&0&e&f&g\\0&0&0&h&i\\0&0&0&0&j\\0&0&0&0&0\end{pmatrix}}\,$

over a field ${}K$. Show that the fifth power of ${}M$ equals ${}0$, that is,

${}M^{5}=MMMMM=0\,.$

Create a solution