The matrix below shows the results of three baseball teams in a series of competition.

1. Write the points awarded for each result as a column matrix in the order Win, Draw, Lose.

The columns of $R$ are in the order Win, Draw, Lose. Hence the points are:

Win $=3$, Draw $=1$, Lose $=0$.

$P=\begin{pmatrix}3\\1\\0\end{pmatrix}$

1. Set up the matrix product using the given results matrix and the points matrix from part (a).

$RP=\begin{pmatrix}12&5&3\\3&8&7\\9&4&4\end{pmatrix}\begin{pmatrix}3\\1\\0\end{pmatrix}$

2. Evaluate each entry by multiplying each row of $R$ by the column matrix $P$.

$RP=\begin{pmatrix}12(3)+5(1)+3(0)\\3(3)+8(1)+7(0)\\9(3)+4(1)+4(0)\end{pmatrix}=\begin{pmatrix}41\\17\\31\end{pmatrix}$

1. Interpret the entries of $RP$ and identify the greatest total.

From part (b),
$RP=\begin{pmatrix}41\\17\\31\end{pmatrix}.$
These are the total points for Gratitude, Respect and Compassion respectively.

The highest number of points is $41$, so the winning team is Gratitude.