Must an idempotent matrix be symmetric?

Question

A matrix $A$ is idempotent if:

$$AA = A$$

Is it true that all such matrices are symmetric?

Answer 1

No. Here's a simple $2 x 2$ counterexample:

Define:

$$A = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix}$$

Note that $A$ is not symmetric, i.e. $$A^T = \begin{pmatrix} 1 &1\\0 &0 \end{pmatrix} \neq A$$

However, $A$ is idempotent:

$$AA = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = \begin{pmatrix} 1 &0\\1 &0 \end{pmatrix} = A$$

Therefore, it cannot be the case that an idempotent matrix has to be symmetric.