Projector equal to identity matrix

Question

I have a simple question. If I take a projector f with matrix $M$, then I have :

$$M^{2}=M$$, so by multiplying by $M^{-1}$ on the right, I get :

$MMM^{-1}= I_{d}$, so $M=I_{d}$, but we can find matrices $M$ which are not identity and represent a projector.

Where is my error ?

Regards

Answer 1

Your error lies in assuming that $M$ is invertible. Actually, you proved that the only invertible projection is the identity.

Answer 2

If $M^2=M$, then we have:

$M$ is invertible $ \iff M=Id$.

Hence (if $M^2=M$), the

$M$ is not invertible $ \iff M \ne Id$.

Try also to prove: if $M^2=M$, then the only possible eigenvalues of $M$ are $0$ and $1$.