If A is a nxn matrix that is idempotent and invertible, then $\displaystyle A=I_{n}$.

Let A be any nxn matrix such that A is idempotent and invertible. Then, by definition A satisfies the following properties,

$\displaystyle A^{2}=A$ and A can be written as the product of elementary matrices.

So, since $\displaystyle A^{2}=A$ we can say $\displaystyle A^{2}=E_{k}^{-1}...E_{1}^{-1}$

...I don't really know how to finish this...

Should I use an equivalent condition for invertible matrices?