The table below shows a multiple-choice test Sam and Roger took.

1. Arrange the table values into a $2\times3$ matrix in the given order.

Rows: Sam, Roger.
Columns: Correct, No attempt, Incorrect.

$S=\begin{pmatrix}14&5&1\\15&0&5\end{pmatrix}$

1. Represent the marks awarded as a $3\times1$ matrix in the correct order.

Since the columns of $S$ are in the order Correct, No attempt, Incorrect,
$T=\begin{pmatrix}2\\0\\-1\end{pmatrix}$

2. Multiply $S$ by $T$ to find the total marks for each contestant.

Using $S=\begin{pmatrix}14&5&1\\15&0&5\end{pmatrix},$
$ST=\begin{pmatrix}14&5&1\\15&0&5\end{pmatrix}\begin{pmatrix}2\\0\\-1\end{pmatrix}$
$=\begin{pmatrix}14(2)+5(0)+1(-1)\\15(2)+0(0)+5(-1)\end{pmatrix}$
$=\begin{pmatrix}27\\25\end{pmatrix}$

1. State what the product matrix $ST$ means in the context of the test scores.

From part (b),
$ST=\begin{pmatrix}27\\25\end{pmatrix}.$
The first row corresponds to Sam and the second row corresponds to Roger. Hence the values are their total marks in the test.