quite nice problem

harazi

If $A_1, A_2,...,A_n$ are nxn complex nilpotent matrices such that any two commute, than their product is the zero matrix.

**Posted:** Wed Oct 27, 2004 12:43 pm

grobber

It was posted a very long time ago here.

**Posted:** Wed Oct 27, 2004 1:36 pm

harazi

Sorry, I did not know that. Anyway, I have a solution which does not use rank or endomorphisms, just a very nice identity. Sorry again for posting this.

**Posted:** Wed Oct 27, 2004 11:59 pm

Moubinool

harazi

Could you post this very nice identity ?

thx

**Posted:** Thu Oct 28, 2004 1:51 am

harazi

The identity is:
$(x_1+x_2+...+x_n)^n-\sum {(x_2+...+x_n)^n}+\sum {(x_3+...+x_n)^n}-...=n! x_1\cdot x_2\cdot...\cdot x_n$ .
Here the sum $\sum {(x_2+...+x_n)^n}$ has n terms, the second one has $\frac{n(n-1)}{2}$ and so on.

**Posted:** Thu Oct 28, 2004 3:10 am