1. **State the problem:** Given matrices representing points per outcome and match results for two teams, we want to find the product matrix $RP$. 2. **Define the points matrix $P$: an array representing points for wins, draws, and losses.** Since wins = 2 points, draws = 1 point, losses = 0 points, and $P$ is a $3 \times 1$ matrix, we have: $$P = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$$ 3. **Define matrix $R$: a $2 \times 3$ matrix showing the results for the Warriors and Spartans.** The problem states the Warriors won 6, drew 3, lost 2; Spartans won 4, drew 2, lost 5. So: $$R = \begin{bmatrix} 6 & 3 & 2 \\ 4 & 2 & 5 \end{bmatrix}$$ 4. **Calculate the product matrix $RP$:** multiply the $2 \times 3$ matrix $R$ by the $3 \times 1$ matrix $P$: $$RP = R \times P = \begin{bmatrix} 6 & 3 & 2 \\ 4 & 2 & 5 \end{bmatrix} \times \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$$ 5. **Perform matrix multiplication step-by-step:** - For the Warriors (first row): $$6 \times 2 + 3 \times 1 + 2 \times 0 = 12 + 3 + 0 = 15$$ - For the Spartans (second row): $$4 \times 2 + 2 \times 1 + 5 \times 0 = 8 + 2 + 0 = 10$$ 6. **Final matrix:** $$RP = \begin{bmatrix} 15 \\ 10 \end{bmatrix}$$ 7. **Interpretation:** This means the Warriors earned 15 points and the Spartans earned 10 points in total from their respective matches.