Square of Elementary Matrix Proof

Question

I'm having trouble proving the following statement:

"There exists an elementary matrix $E_1$ such that $E_1^2 = I$"

I'm thinking about how the inverse of $E_1$ is equal to $E_1$ (so $E_1^{-1} = E_1$) but I'm not sure how to show the product of it, if that is even the right step towards the proof. Could someone help me out?

Answer 1

Forget matrices. An elementary matrix "does something" - multiplying by $E$ swaps two rows or multiplies a row by a constant or adds a multiple of this row to that row.

Now think about finding one of those somethings with this property: If you do it twice you get back to where you started...

Answer 2

Hint: In order to prove that such a matrix exists, find an example. Try to think of a row-operation that, when applied twice (in a row), gives you what you started with.