Mrs Teo is a Mathematics tutor. She offers tutorial sessions for Lower Secondary and Upper Secondary students on weekdays and on weekends. Each student attends one session a week for 4 sessions in every month. The matrix N shows the number of students she tutors every month. Columns correspond to Lower Secondary (first column) and Upper Secondary (second column). Rows correspond to Weekday (first row) and Weekend (second row). $N = \begin{pmatrix}7 & 6 \\\ 4 & 3\end{pmatrix}$

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:
$1747\times 12=20964$