一家旅行社销售去往越南、马尔代夫和台湾的邮轮套餐，价格分别为 $1299、$1398 和 $2538。下表显示了1月至6月以及7月至12月期间，通过该旅行社报名参加各邮轮套餐的客户人数。 报名参加越南、马尔代夫和台湾邮轮套餐的人数可以用矩阵表示为 $P=\begin{pmatrix}27 & 24\\23 & 29\\19 & 22\end{pmatrix}$

1. Write the price matrix in the correct order and dimensions for $C=AP$.

Since $P$ is a $3\times 2$ matrix with rows in the order Vietnam, Maldives, Taiwan, and $C$ must give total earnings for each period, $A$ must be a $1\times 3$ row matrix containing the corresponding prices:
$A=\begin{pmatrix}1299 & 1398 & 2538\end{pmatrix}.$

1. Set up the matrix multiplication $C=AP$ using the price matrix and the customer matrix.

$C=AP$
$=\begin{pmatrix}1299 & 1398 & 2538\end{pmatrix}\begin{pmatrix}27 & 24\\23 & 29\\19 & 22\end{pmatrix}$

2. Multiply corresponding entries and add to find each period's total earnings.

$C=\begin{pmatrix}(1299)(27)+(1398)(23)+(2538)(19) & (1299)(24)+(1398)(29)+(2538)(22)\end{pmatrix}$
$=\begin{pmatrix}35073+32154+48222 & 31176+40542+55836\end{pmatrix}$
$=\begin{pmatrix}115449 & 127554\end{pmatrix}$

1. Set up the matrix multiplication $M=CR$ using the given column matrix $R$.

$M=CR=\begin{pmatrix}118\,911 & 127\,584\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}$

2. Multiply corresponding entries and add to obtain the single element of $M$.

$M=\begin{pmatrix}118\,911(1)+127\,584(1)\end{pmatrix}$

$M=\begin{pmatrix}246\,495\end{pmatrix}$

1. Interpret the single element of $M$ in the context of the question.

Since $M=CR=\begin{pmatrix}118\,911 & 127\,584\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix},$ the two period earnings are added together. Hence, the element of $M$ represents the total earnings for both periods combined, from January to December.