罗伯特在每个工作日的早晨有 3 种上学方式的选择。 方案 A：他乘公共汽车。 方案 B：他乘地铁（MRT）。 方案 C：他乘出租车。 矩阵 A、B 和 C 分别代表他如果选择方案 A、B 或 C 时上学的路程。

1. Multiply matrix $A$ by 3.

$3A=3\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}=\begin{pmatrix}3&0&0\\0&0&0\\0&0&0\end{pmatrix}$

2. Add the matrices entry by entry to find $S$.

$S=3A+B+C$
$=\begin{pmatrix}3&0&0\\0&0&0\\0&0&0\end{pmatrix}+\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}+\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}$
$=\begin{pmatrix}3&0&0\\0&1&0\\0&0&1\end{pmatrix}$

1. Write the given travelling times as a column matrix in the order Bus, MRT, Taxi.

The times are Bus $=30$, MRT $=20$, Taxi $=15$.

Hence,
$N=\begin{pmatrix}30\\20\\15\end{pmatrix}$

1. Set up the matrix product using the answers from earlier parts.

From earlier parts,
$S=\begin{pmatrix}3&0&0\\0&1&0\\0&0&1\end{pmatrix},\qquad N=\begin{pmatrix}30\\20\\15\end{pmatrix}.$
Hence
$SN=\begin{pmatrix}3&0&0\\0&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}30\\20\\15\end{pmatrix}.$

2. Multiply each row of $S$ by the column matrix $N$.

$SN=\begin{pmatrix}3(30)+0(20)+0(15)\\0(30)+1(20)+0(15)\\0(30)+0(20)+1(15)\end{pmatrix}$
$=\begin{pmatrix}90\\20\\15\end{pmatrix}.$

1. Interpret the entries of $SN$ in the context of the journey options.

Since $SN=\begin{pmatrix}90\\20\\15\end{pmatrix},$ the entries represent the total travelling time for the week by each option, in the order Bus, MRT, Taxi.

So Robert spent $90$ minutes travelling by bus, $20$ minutes by MRT and $15$ minutes by taxi.

1. Write a row matrix that averages the three total-time entries in $SN$ over 5 weekdays.

Since $SN=\begin{pmatrix}90\\20\\15\end{pmatrix}$ contains the total travelling times for the 3 options in the week, the average per day is
$\frac{90+20+15}{5}.$
Thus the matrix multiplying $SN$ should be
$J=\begin{pmatrix}\dfrac15&\dfrac15&\dfrac15\end{pmatrix}.$

1. Use matrix multiplication to find the weekly average travelling time.

From earlier parts,
$J=\begin{pmatrix}\dfrac15&\dfrac15&\dfrac15\end{pmatrix},\quad SN=\begin{pmatrix}90\\20\\15\end{pmatrix}.$
Hence,
$J(SN)=\begin{pmatrix}\dfrac15&\dfrac15&\dfrac15\end{pmatrix}\begin{pmatrix}90\\20\\15\end{pmatrix}$
$=\dfrac15(90)+\dfrac15(20)+\dfrac15(15)$

2. Calculate and state the average with units.

$\dfrac{90+20+15}{5}=\dfrac{125}{5}=25$
So the average travelling time per day is $25$ minutes.