Is the set of all Idempotent matrix in $M_n(\mathbb{F})$ linearly independent?

Question

Is the set $I:=\{\text{set of all Idempotent matrix in}\ M_n(\mathbb{F})\}$ linearly independent?

My thought: I think the answer is no if $\mathbb{F}$ is infinite.
If $\mathbb{F}$ is infinite then the class of all idempotent matrix is infinite. Any subset of $I$ with more than $n^2$ elements is not linearly independent [as dim($M_n(\mathbb{F}))=n^2$]. As $\mathbb{F}$ is infinite hence $I$ is infinite.

But what about if $\mathbb{F}$ is finite?

Answer 1

Hint If $A$ is idempotent then so is $I-A$. Moreover, $I$ is idempotent.

Note The problem contains a small trap, you need to make sure that there exists at least one other idempotent matrix besides $I$. This happens excepting when $n=..$?

Answer 2

Of course not: $\begin{bmatrix}1&0\\0&0\end{bmatrix}+ \begin{bmatrix}0&0\\0&1\end{bmatrix}= \begin{bmatrix}1&0\\0&1\end{bmatrix}$ for any ring used as coefficients.

Answer 3

$O^2=O$, and so is idempotent. No set containing the zero "vector" is linearly independent.