1. **State the problem:** Multiply the 2x2 matrix $$\begin{bmatrix}1 & 0 \\ 3 & -1\end{bmatrix}$$ by the 2x1 vector $$\begin{bmatrix}2 \\ 1\end{bmatrix}$$. 2. **Recall the matrix multiplication rule:** To multiply a matrix $$A$$ by a vector $$\mathbf{x}$$, multiply each row of $$A$$ by the vector $$\mathbf{x}$$ and sum the products. 3. **Perform the multiplication:** - First row: $$1 \times 2 + 0 \times 1 = 2 + 0 = 2$$ - Second row: $$3 \times 2 + (-1) \times 1 = 6 - 1 = 5$$ 4. **Write the resulting vector:** $$\begin{bmatrix}2 \\ 5\end{bmatrix}$$. 5. **Match with the given options:** Option b is $$\begin{bmatrix}2 \\ 5 \\ 3\end{bmatrix}$$ (3x1 vector), which is not correct. Option d is $$\begin{bmatrix}2 \\ 7\end{bmatrix}$$, which is incorrect. Option a is $$[7]$$, single element, incorrect. Option c is $$[5]$$, single element, incorrect. Option e is a 2x3 matrix, incorrect. 6. The correct answer is the vector $$\begin{bmatrix}2 \\ 5\end{bmatrix}$$, but it is not exactly listed. Since option b has the first two elements correct but an extra 3, none match exactly. **Final answer:** $$\begin{bmatrix}2 \\ 5\end{bmatrix}$$ (closest to option b but without the extra 3).