Skew Symmetric Matrix with condition

Question

If A is 2*2 skew symmetric matrix with $$A^2=A$$, Then A=0 $$$$ My attempt was by assuming the matrix A and substituting the squaring condition and the condition of skew symmetric and got system of equations but I couldn't solve it.I look forward to your comments.

Answer 1

$$A=A^2=(-A)^2=(A^T)^2=A^TA^T=(AA)^T=A^T=-A$$

This is valid in all cases where $\mathrm{char}(F)\ne2$. It can be easily prooved by defenition, and does not matter on size of $A$.

Answer 2

Let$$A=\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}$$

Then $A^2=A$ implies

$$\begin{bmatrix} -a^2 & 0 \\ 0 & -a^2 \end{bmatrix}=\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}$$

Hence $a=0$ and $A=O$