On Wikipedia, I found that the only non-singular idempotent matrix is $\mathrm{\mathbf{I}}$. However, the following matrix
$$\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0 \\ 0 & 1 & 1\end{pmatrix}$$
is invertible and is equal to itself when squared. What is it that I am misunderstanding?
If $A^2= A$, and $A$ is invertible, then we can multiply both sides by the inverse of $A$, and we get $A=I$. That reasoning seems rock solid, to me.
Are you sure that the matrix you showed is both idempotent and invertible?