Giving an example of a nilpotent matrix

Hi MHF,

The exercise states "Give an example of a 3x3 nilpontent matrix which is not the null matrix".

My attempt : I wrote down the matrix $\displaystyle

\begin{bmatrix}

a & b & c\\

d & e & f\\

g & h & i\end{bmatrix}

$

and I multiplied it by itself. I got the matrix $\displaystyle

\begin{bmatrix}

a^2+bd+cg & ab+be+ch & ac+bf+ic\\

ad+ed+fg & bd+e^2+fh & cd+ef+if\\

ag+eh+ig & bg+eh+ih & cg+hf+i^2 \end{bmatrix}

$

Now I must equal each entriy of the matrix to 0... that means I'll get a "nasty" system of 9 equations with 9 unknown to solve. I say "nasty" because it isn't even linear!

So I'd like to know what's the method to find such an example. Without guessing of course. I don't think my method is effective.

Thank you.