Linear Algebra — verifying g-inverse properties for given matrices

For the matrices $A = \begin{bmatrix} 4 & 2 & 2 \ 2 & 2 & 0 \ 2 & 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 & 0 \ 0 & \frac{1}{2} & 0 \ 0 & 0 & \frac{1}{2} \end{bmatrix}$, consider the following statements: I. $ABA = A$ II. $BA$ is an idempotent matrix. III. Rank $(A) \ne$ Rank $(BA)$ IV. Rank $(BA)$ = Trace $(BA)$ Select the correct answer using the code given below.