1. **Problem statement:** Calculate the combined transformation matrix and the image of point $P(3,-2)$ after two successive transformations given by matrices $$A = \begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}$$ and $$B = \begin{bmatrix}0 & 2 \\ 1 & 0\end{bmatrix}$$ 2. **Formula and rules:** The combined transformation matrix for two transformations $A$ followed by $B$ is given by matrix multiplication: $$C = B \times A$$ Matrix multiplication is done by taking the dot product of rows of the first matrix with columns of the second matrix. 3. **Calculate the combined matrix $C$: ** $$C = \begin{bmatrix}0 & 2 \\ 1 & 0\end{bmatrix} \times \begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}0 \times 1 + 2 \times 0 & 0 \times (-1) + 2 \times 1 \\ 1 \times 1 + 0 \times 0 & 1 \times (-1) + 0 \times 1\end{bmatrix} = \begin{bmatrix}0 & 2 \\ 1 & -1\end{bmatrix}$$ 4. **Find the image of point $P(3,-2)$:** Represent $P$ as a column vector: $$\mathbf{p} = \begin{bmatrix}3 \\ -2\end{bmatrix}$$ Apply the combined transformation: $$\mathbf{p'} = C \times \mathbf{p} = \begin{bmatrix}0 & 2 \\ 1 & -1\end{bmatrix} \times \begin{bmatrix}3 \\ -2\end{bmatrix} = \begin{bmatrix}0 \times 3 + 2 \times (-2) \\ 1 \times 3 + (-1) \times (-2)\end{bmatrix} = \begin{bmatrix}-4 \\ 5\end{bmatrix}$$ 5. **Answer:** (i) The combined transformation matrix is $$\boxed{\begin{bmatrix}0 & 2 \\ 1 & -1\end{bmatrix}}$$ (ii) The image of point $P(3,-2)$ under the combined transformation has coordinates $$\boxed{(-4, 5)}$$