Find the square matrix $A$ that satisfies $2A^T=A+\begin{bmatrix}1&0\\0&-1\end{bmatrix}$. **(i)** $\begin{bmatrix}1&1\\0&-1\end{bmatrix}$ **(ii)** $\begin{bmatrix}1&0\\-1&1\end{bmatrix}$ **(iii)** $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$