The table shows the passes sold on two festival days. Day pass Weekend pass Saturday 120 45 Sunday 90 30 Write the sales as a 2 by 2 matrix. (1) A day pass ...
Question 1 Report
The table shows the passes sold on two festival days.
| Day pass | Weekend pass | |
|---|---|---|
| Saturday | 120 | 45 |
| Sunday | 90 | 30 |
- Write the sales as a 2 by 2 matrix. (1)
- A day pass is £30 and a weekend pass £80. Multiply your matrix by \(\begin{pmatrix} 30 \\ 80 \end{pmatrix}\) and say what the entries mean. (2)
Answer Details
(a) Keeping the rows as the days and the columns as the pass types, exactly as the table is laid out:
\[\begin{pmatrix} 120 & 45 \\ 90 & 30 \end{pmatrix}\][1]
(b) In matrix multiplication each entry of the answer comes from one row of the first matrix and one column of the second: multiply matching entries and add. The column matrix \(\begin{pmatrix} 30 \\ 80 \end{pmatrix}\) holds the price of a day pass above the price of a weekend pass, in the same order as the columns of the sales matrix, which is what makes the multiplication meaningful.
\[\begin{pmatrix} 120 & 45 \\ 90 & 30 \end{pmatrix}\begin{pmatrix} 30 \\ 80 \end{pmatrix}\]First row: \(120 \times 30 + 45 \times 80 = 3600 + 3600 = 7200\).
Second row: \(90 \times 30 + 30 \times 80 = 2700 + 2400 = 5100\).
\[= \begin{pmatrix} 7200 \\ 5100 \end{pmatrix}\][1]
The entries are the money taken in pounds: £7200 on Saturday and £5100 on Sunday. [1]
Each entry is a number of passes multiplied by a price per pass, so the units work out as pounds, and each row adds both kinds of pass for one day. A \(2 \times 2\) matrix multiplied by a \(2 \times 1\) matrix gives a \(2 \times 1\) result, which is why the answer has one figure per day rather than one per pass type.
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