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

and I multiplied it by itself. I got the matrix

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.