IB Maths AI HL AI 1.5 Matrices questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

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Question 1

Paper 1Medium5 marks

A shop records the weekly sales of two products. The sales are shown in Figure 1. The selling prices remain constant: product A costs $4.80 per unit and product B costs $6.25 per unit.

(a)

Write down the $2\times3$ matrix $S$ of weekly sales, with rows corresponding to products A and B, in that order, and columns corresponding to Weeks 1, 2 and 3, in that order.

[2]

(b)

Let $\mathbf p=\begin{bmatrix}4.80&6.25\end{bmatrix}$ the row matrix of prices in dollars per unit.

Using your matrix from part (a), show that the revenue in Weeks 1, 2 and 3 is given by

\mathbf pS=\begin{bmatrix}215.20&219.00&261.40\end{bmatrix}.

[2]

(c)

Hence, state the week with the greatest revenue.

[1]

A field station sends four-letter locker identifiers using a linear code. The letters A to Z are represented by the integers 1 to 26, respectively. Each successive pair of letters is written as a column vector $\mathbf{x}$ and is encoded as $\mathbf{y}=K\mathbf{x}$

The transmitted pairs, in order, are

\begin{pmatrix}27\\14\end{pmatrix} \quad\text{and}\quad \begin{pmatrix}48\\28\end{pmatrix}.

The same encoding matrix was used for both pairs. It was one of

P=\begin{pmatrix}2&1\\1&1\end{pmatrix} \quad\text{or}\quad Q=\begin{pmatrix}1&2\\1&3\end{pmatrix}.

(a)

Show that $P^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}$

[2]

(b)

Determine which matrix was used and hence decode the locker identifier. Justify your answer.

[4]

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