Show that, if all the entries of $A$ on and below the diagonal are zero, then $A$ is nilpotent.
I know this has been asked before , but I want to solve this question without the use of Cayley-Hamilton theorem, which I am able to do.
All I know so far is it has eigenvalue $0$ with algebraic multiplicity $n$. Any hints on how to proceed would be appreciated, thanks.
I also notice that a matrix with the above form size $2\times2$ and $3\times3$ has order $2$ and $3$, respectively.
Is there a way to show for an $m \times m$ matrix that is $A^m=(0)$?