If $\displaystyle A$ is a $\displaystyle n \times n$ matrix, and satisfy $\displaystyle A^{2} = A$.

show that: $\displaystyle rank(A) = tr(A)$

vice verse, If matrix $\displaystyle A$ satisfy the condition : $\displaystyle rank(A) = tr(A)$ , $\displaystyle (*)$

then $\displaystyle A$ not always Idempotent matrix, such as: $\displaystyle \left(\begin{array}{cc}1 & 2 \\

1 & 1\end{array}\right)$(corrected)

can we add some conditions to $\displaystyle "(*)"$ such that $\displaystyle A$ is a Idempotent matrix