Counting idempotent $2×2$ matrices over $\mathbb{R}$

Question

Question: if $A$ is a real $2×2$ matrix such that $A^2 - A= 0$ then which of the following are true?

(a) there are infinitely many such matrices $A$

(b) there are finitely many such matrices $A$

(C) $A$ has to be diagonal matrix.

Please help me. I know that, the minimal polynomial must divides the annihilating polynomial of matrix. Hence possibilities of minimal polynomial are: $m(x) =x$ or $m(x) = x-1$ or $m(x) =x(x-1)$ but how to determine number of matrices from this?

Answer 1

Consider the matrices $MDM^{-1}$ where $M$ is any $2\times 2$ invertible matrix and $$D=\begin{pmatrix}0&0\\0&1\end{pmatrix}.$$ P.S. Explicit example: for $\theta\in [0,\pi),$ $$A=\begin{pmatrix}\sin^2(\theta)&-\sin(\theta)\cos(\theta)\\-\sin(\theta)\cos(\theta)&\cos^2(\theta)\end{pmatrix}$$

Answer 2

The matrix of the orthogonal projection onto a 1-dimensional subspace of $\mathbb R^2$ is idempotent. Since there are infinitely many such subspaces, there are infinitely many idempotent matrices.

Actually, this argument proves that there are infinitely many similarity classes of idempotent matrices.