1. **State the problem:** We have a parallelogram represented by the matrix $$\begin{bmatrix}1 & 2 \\ 3 & 5 \end{bmatrix}$$ which is transformed by the matrix $$\begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix}$$. We want to find in which quadrant the transformed parallelogram lies. 2. **Understand the transformation:** The matrix $$\begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix}$$ reflects points across the x-axis by negating the y-coordinates. 3. **Apply the transformation:** Multiply the transformation matrix by the original matrix: $$\begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix}1 & 2 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix}1 \times 1 + 0 \times 3 & 1 \times 2 + 0 \times 5 \\ 0 \times 1 + (-1) \times 3 & 0 \times 2 + (-1) \times 5 \end{bmatrix} = \begin{bmatrix}1 & 2 \\ -3 & -5 \end{bmatrix}$$ 4. **Interpret the result:** The transformed parallelogram has vertices with x-coordinates positive (1, 2) and y-coordinates negative (-3, -5). 5. **Determine the quadrant:** Points with positive x and negative y lie in Quadrant IV. **Final answer:** The transformed parallelogram lies in Quadrant IV. **Answer choice:** B. IV