The matrix T shows the number of training sessions Alyssa and Farah attended for the different training programmes in a year. Circuit Interval Long Run $T=\begin{pmatrix}50 & 100 & 150 \\ 60 & 100 & 160\end{pmatrix}$

1. Arrange the durations in the required column matrix order

The programmes are ordered as Circuit, Interval, Long Run. Their durations are $40$, $15$ and $120$ minutes respectively.

Since $S$ is a $3\times 1$ column matrix,
$S=\begin{pmatrix}40\\15\\120\end{pmatrix}.$

1. Multiply the $2\times 3$ matrix $T$ by the $3\times 1$ matrix $S$.

Using $T=\begin{pmatrix}50&100&150\\60&100&160\end{pmatrix},\quad S=\begin{pmatrix}40\\15\\120\end{pmatrix},$
$R=TS=\begin{pmatrix}50&100&150\\60&100&160\end{pmatrix}\begin{pmatrix}40\\15\\120\end{pmatrix}$
$=\begin{pmatrix}50(40)+100(15)+150(120)\\60(40)+100(15)+160(120)\end{pmatrix}$
$=\begin{pmatrix}21500\\23600\end{pmatrix}.$

1. Interpret each element of $R$ in the context of the matrix multiplication.

Since the first row of $T$ is for Alyssa and the second row is for Farah, and $S$ gives the durations in minutes, the elements of
$R=\begin{pmatrix}21500\\23600\end{pmatrix}$
represent the total training time in minutes for Alyssa and Farah respectively.

1. Substitute $R$ and multiply the matrices

From part (b),
$R=\begin{pmatrix}21500\\23600\end{pmatrix}.$
Hence
$P=\begin{pmatrix}-1&1\end{pmatrix}\begin{pmatrix}21500\\23600\end{pmatrix}$
$=(-1)(21500)+(1)(23600)=2100.$

1. Interpret the matrix operation and state its contextual meaning

Since $R=\begin{pmatrix}21500\\23600\end{pmatrix}$ gives Alyssa's and Farah's total training durations respectively,
$P=\begin{pmatrix}-1&1\end{pmatrix}R=-\text{(Alyssa's total duration)}+\text{(Farah's total duration)}.$
Thus $P=2100$ represents the difference in total training duration: Farah trained $2100$ minutes more than Alyssa.