张太太（Mrs Teo）是一名数学补习老师。 她在工作日和周末为初中低年级（Lower Secondary）和初中高年级（Upper Secondary）的学生提供补习课程。 每个学生每周上一次课，每月共上 4 次课。 矩阵 N 表示她每月辅导的学生人数。 列对应于初中低年级（第一列）和初中高年级（第二列）。行对应于工作日（第一行）和周末（第二行）。 $N = \begin{pmatrix}7 & 6 \\\ 4 & 3\end{pmatrix}$

1. Multiply each entry of $N$ by $4$.

$M=4N=4\begin{pmatrix}7&6\\4&3\end{pmatrix}=\begin{pmatrix}4(7)&4(6)\\4(4)&4(3)\end{pmatrix}=\begin{pmatrix}28&24\\16&12\end{pmatrix}$

1. Place the Lower Secondary and Upper Secondary charges into a $2\times1$ column matrix in the correct order.

Since the columns of $N$ are ordered as Lower Secondary, then Upper Secondary,
$C=\begin{pmatrix}\text{Lower Secondary charge}\\\text{Upper Secondary charge}\end{pmatrix}=\begin{pmatrix}70\\80\end{pmatrix}.$

1. Multiply the $2\times2$ matrix $M$ by the $2\times1$ matrix $C$.

Using
$M=\begin{pmatrix}28&24\\16&12\end{pmatrix},\qquad C=\begin{pmatrix}70\\80\end{pmatrix},$
$P=MC=\begin{pmatrix}28&24\\16&12\end{pmatrix}\begin{pmatrix}70\\80\end{pmatrix}$
$=\begin{pmatrix}28(70)+24(80)\\16(70)+12(80)\end{pmatrix}$
$=\begin{pmatrix}1960+1920\\1120+960\end{pmatrix}$
$=\begin{pmatrix}3880\\2080\end{pmatrix}.$

1. Interpret each element of $P$ according to the row categories after multiplying the number of monthly sessions by the session charges.

Since $M$ gives the total number of sessions per month for each category of student, and $C$ gives the charge per session, $P=MC=\begin{pmatrix}3880\\2080\end{pmatrix}$ gives the monthly income from the two rows: weekday sessions and weekend sessions respectively.

1. Find the weekend session charges after a 10% increase.

Weekend Lower Secondary charge:
$70\times 1.10=77$
Weekend Upper Secondary charge:
$80\times 1.10=88$

2. Calculate the total earnings for one session in a week from all registered students.

Weekday earnings per weekly session:
$10(70)+8(80)=700+640=1340$
Weekend earnings per weekly session:
$3(77)+2(88)=231+176=407$
Total earnings per week:
$1340+407=1747$

3. Convert the weekly earnings to total earnings over 3 months.

Each student attends $4$ sessions every month.

For $3$ months, number of weekly sessions per student is
$4\times 3=12$
Total amount earned:
$1747\times 12=20964$