Basis for the vector space of idempotents

Homework Statement

A matrix $$a$$ is idempotent if $$a^2=a$$. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents

2. The attempt at a solution
the vector space in question is dimension 4, so I need to find 4 idempotent matrices.
but i don't want to find them by 'guessing'. Is there a good way to find them?

Thanks

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Mark44
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Homework Statement

A matrix $$a$$ is idempotent if $$a^2=a$$. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents

2. The attempt at a solution
the vector space in question is dimension 4, so I need to find 4 idempotent matrices.
but i don't want to find them by 'guessing'. Is there a good way to find them?
I'm guessing that the dimension of the subspace of idempotent 2 x 2 matrices is less than 4.

I would work with the equation A2 = A, and see what I could come up with about a matrix in this space.

It might also be helpful to look at an arbitrary matrix, such as
$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Note that in every text I've ever seen, matrices are represented by capital letters; e.g., A, B, C, and so on.