1.14 Matrices questions for IB Maths AI

Write down the order of $\mathit{S}$.

Write down the element $s_{23}$ and state what it represents.

Find .

Interpret the second element of in context.

The sprite is mapped by $z\mapsto \mathrm{i}z$. Find the images of $A$ and $C$.

Describe fully the single transformation $z\mapsto \mathrm{i}z$.

By writing $z=x+y\mathrm{i}$, show that the transformation $z\mapsto (a+b\mathrm{i})z$, where $a,b\in \mathbb{R}$, can be represented by the matrix

The sprite is now mapped by $z\mapsto (1+\mathrm{i})z$. Find the image of $C$.

Find the area of triangle $ABC$ and hence find the area of its image under $z\mapsto (1+\mathrm{i})z$.

Mei wants a single transformation $z\mapsto wz$ that maps $A$ onto $B$. Find $w$ in the form $a+b\mathrm{i}$.

(i) Find the image of $C$ under the transformation in part (f).

(ii) Hence state whether the image of the sprite is congruent to triangle $ABC$. Give a reason for your answer.

Write down the matrix $\mathit{L}$ such that $\left(\begin{array}{c}{J}_{n+1}\\ A_{n+1}\end{array}\right)=\mathit{L}\left(\begin{array}{c}{J}_n\\ A_n\end{array}\right)$.

(i) Find $J_1$ and $A_1$.

(ii) Find the total number of females in the reserve after 6 years, according to this model.

Show that the eigenvalues of $\mathit{L}$ are $1.1$ and $-0.3$.

Find an eigenvector of $\mathit{L}$ for each eigenvalue.

Show that $\left(\begin{array}{c}310\\ 130\end{array}\right)=50\left(\begin{array}{c}5\\ 3\end{array}\right)+20\left(\begin{array}{c}3\\ -1\end{array}\right)$.

Hence show that the total number of females after $n$ years is $T_n=400(1.1{)}^n+40(-0.3{)}^n$.

Find the first year in which the total number of females is predicted to exceed 3000.

Describe the long-term behaviour of the beaver population according to this model. Justify your answer.

The reserve cannot support an ever-growing population, so the ecologists plan to relocate some adult females each year. This changes only the proportion of adult females that remain as adults in the reserve, from 0.6 to $a$.

The population is stable in the long term when $\mathit{L}$ has an eigenvalue equal to 1.

Find the value of $a$ for which the population is stable.

The natural survival rate of adults remains 60 %, and the relocated females are chosen from the adults who would otherwise have remained.

Find the percentage of these adult females that must be relocated each year.

Explain what the model predicts if the ecologists relocate slightly more than this percentage every year.

State two limitations of this model.

Find the value of $x$, the value of $y$ and the value of $z$.

Using your values of $x$ and $y$, find .

Find ${\mathit{W}}_1+{\mathit{W}}_2$.

State what the element in row 2, column 2 of ${\mathit{W}}_1+{\mathit{W}}_2$ represents.

In week 3, the sales of every item at both shops were 20% higher than in week 1.

Find the matrix ${\mathit{W}}_3$ for week 3.

The selling prices, in British pounds (GBP), are 180 for a tent, 45 for a sleeping bag and 32 for a stove. They are written as the column vector $\mathit{p}$.

Find $({\mathit{W}}_1+{\mathit{W}}_2)\mathit{p}$ and explain what it represents.

Kestrel Outdoors buys a tent for 110 GBP, a sleeping bag for 28 GBP and a stove for 19 GBP. These costs are written as the column vector $\mathit{c}$.

By finding a suitable matrix product, find the profit made by each shop over weeks 1 and 2.

Find $\left(\begin{array}{cc}1& 1\end{array}\right)({\mathit{W}}_1+{\mathit{W}}_2)$ and interpret your answer.

Explain why the product $\mathit{p}({\mathit{W}}_1+{\mathit{W}}_2)$ is not defined.

At one moment there are 6000 aphids and 100 ladybirds in the greenhouse.

Find the value of $\frac{\mathrm{d}x}{\mathrm{d}t}$ and the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at this moment, and interpret your answers in context.

Consider the case in which there are no ladybirds in the greenhouse.

Write down the differential equation satisfied by $x$.

Hence find the long-term number of aphids when there are no ladybirds, justifying your answer.

Show that $(4,2)$ is an equilibrium point of the system.

Write down the coordinates of the other two equilibrium points.

To investigate the behaviour close to $(4,2)$, let $x=4+p$ and $y=2+q$, where $p$ and $q$ are small. You may assume that terms in $p^2$, $q^2$ and $pq$ can be ignored.

Show that $\frac{\mathrm{d}p}{\mathrm{d}t}\approx -0.4p-1.6q$.

Find a similar approximation for $\frac{\mathrm{d}q}{\mathrm{d}t}$.

The approximations can be written as $\left(\begin{array}{c}\dot{p}\\ \dot{q}\end{array}\right)=\mathit{M}\left(\begin{array}{c}p\\ q\end{array}\right)$.

Find the eigenvalues of $\mathit{M}$.

Hence describe the behaviour of the aphid and ladybird populations when they start close to $(4,2)$.

At the start of one season there are 2000 aphids and 100 ladybirds in the greenhouse.

Use Euler's method with a step length of 0.25 weeks to estimate the number of aphids and the number of ladybirds after 1 week.

Sketch, on a set of axes with $x$ on the horizontal axis and $y$ on the vertical axis, the trajectory that starts at $(2,1)$. Show the direction of motion along the trajectory.

Selin considers spraying a pesticide that kills aphids and ladybirds at the same rate. She modifies her model to

Find the equilibrium point of the modified model at which both species are present.

Comment on whether spraying the pesticide is a sensible way to reduce the aphid population in the long term.