IB Mathematics Applications & Interpretation AHL 1.14—introduction to Matrices Questionbank

A loyalty programme converts two balances by the exact rule $y=Ax$, where $x$ is a column of original points in tiers A and B, $y$ is a column of converted credits in channels 1 and 2, and $A=\begin{pmatrix}4 & 3\\2 & 5\end{pmatrix}$. The conversion rates are credits per original point. The auditor receives $y=\begin{pmatrix}126\\112\end{pmatrix}$ and must reverse the conversion, allowing fractional points during the calculation. A database may store a rounded inverse. The audit tolerance is introduced when that storage is tested.

(a) Write down the two scalar conversion equations and explain why adding the two reported credits does not directly give the original total.

(b) Find $\det A$ and $A^{-1}$ by hand.

A transformation is represented by $C = \begin{pmatrix} 5 & p \\ 0 & 5 \end{pmatrix}$. After three applications, its matrix is $\begin{pmatrix} 125 & 750 \\ 0 & 125 \end{pmatrix}$. Find the value of $p$.

Arven Catering buys three ingredients, measured in kilograms, for sites North and South. It must choose one supplier for both sites and certify a revised order against stock and a budget. The quantities and exact unit prices, in euros per kilogram, are:

Let $Q$ have rows North, South and columns rice, beans, vegetables. Let $C$ have rows in that ingredient order and columns Supplier 1, Supplier 2. Transport is included in these rates. Report monetary amounts to two decimal places.

(b) Calculate the matrix of site costs for the two suppliers, and interpret its North, Supplier 2 element.

Let the matrix $B$ be defined as $B=\begin{pmatrix} 2e^u & 4e^{-u} \\ 1-e^u & 3 \end{pmatrix}$, for $u \in \mathbb{R}$.

(d) Determine the first day meeting this criterion, checking the preceding day.

The proposed arrivals change to $b_{\text{new}}=\begin{pmatrix}9\\6\end{pmatrix}$; each queue has capacity $20$ staff-hours. $M$ stays unchanged.

(e) Calculate the revised fixed state and determine whether its two queues meet capacity.

A separate trial uses $s_{n+1}=Ns_n+c$, where $N=\begin{pmatrix}1.1&0\\0&0.5\end{pmatrix}$, $c=\begin{pmatrix}1\\1\end{pmatrix}$ and $s_0=\begin{pmatrix}0\\0\end{pmatrix}$.

(f) Find the alternative fixed state, calculate its state after five days, and compare them.

(g) Contrast the standing of the fixed state as a settled workload under the library's planning rule with its standing under the trial rule, referring to how the queues behave day by day in each case.

(c) Calculate the original balances and their total using the exact inverse.

The database rounds every inverse entry to three decimal places and stores the resulting matrix $R$. In the remaining storage test, use these rounded entries exactly.

(d) Write down the stored matrix $R$ and calculate $AR$ exactly, using the stored entries as actual inputs.

Each recovered balance must be within $0.010$ points, inclusive, of the exact original balance.

(a) Write down $Q$ and $C$, and state their orders.

(c) Calculate the combined costs using a row matrix, and identify the cheaper supplier for the original order.

The North site replaces $5$ kg of vegetables with $5$ kg of rice. South replaces $10$ kg of beans with $10$ kg of vegetables; all other quantities stay unchanged.

(d) Write down the adjustment matrix $D$ and calculate the revised quantities and costs.

Available stock is $75$ kg of rice, $60$ kg of beans and $55$ kg of vegetables.

Determine the real value of $u$ such that the matrix $B$ does not have an inverse.