Matrices (definition, algebra, multiplication, solving systems of equations): notes and practice questions

A celestial mapping transformation $S: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} x' \\ y' \end{pmatrix}$ is defined by

$S: \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 5 & -8 \\ 2 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -7 \\ 3 \end{pmatrix}$.

(a) Find the coordinates of the image of the star system located at (4, -1).

(b) Given that a specific cosmic dust cloud, located at $\begin{pmatrix} p \\ q \end{pmatrix}$, is mapped to $3\begin{pmatrix} p \\ q \end{pmatrix}$ by the transformation $S$, find the value of $p$ and the value of $q$.

(c) A nebula with a certain area in the celestial plane is transformed by $S$. Explain why the transformed nebula will have exactly the same area as the original nebula.

(a) Solve the following matrix equation for $X$:

$2X + \begin{pmatrix} 4 & 1 \\ -2 & 3 \end{pmatrix} = \begin{pmatrix} 10 & 7 \\ 0 & -1 \end{pmatrix}$

(b) Solve the following matrix equation for $X$:

$\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} X = \begin{pmatrix} 7 & -1 \\ 2 & 0 \end{pmatrix}$

(c) Solve the following matrix equation for $X$:

$X \begin{pmatrix} 4 & -1 \\ -3 & 1 \end{pmatrix} - \begin{pmatrix} 0 & 2 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 5 \\ -2 & 3 \end{pmatrix}$

A civil engineer is designing a new road. A preliminary section of the road is modelled by a straight line with equation $y = kx + d$.

Find the vectors $\mathbf{a}$ and $\mathbf{b}$ such that the equation of the line can be expressed in vector form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ in terms of $k$ and/or $d$.

Before construction, a ground transformation is applied to the design. This transformation is described by the matrix $T = \begin{pmatrix} 4 & 2 \\ 6 & 3 \end{pmatrix}$.

Calculate the value of $\det T$.

The preliminary road section $y = kx + d$ (where $k \neq -2$) undergoes the transformation described by matrix $T$.

Show that the equation of the resulting transformed path does not depend on $k$ or $d$.

In a digital art project, points on a canvas are transformed using a matrix operation followed by a translation. The transformation maps an original point $(x, y)$ to its new position $(x', y')$ according to the rule:

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -3 \\ 1 \end{pmatrix}$

(a) Find the new position of a design element originally located at $(4, -2)$.

(b) A specific design feature is observed at the new position $(-1, 10)$. Determine its original coordinates.

(c) An artist uses a series of points $(k, k-1)$ to draw a line segment. Find the general expression for the transformed coordinates of these points in terms of $k$.

A graphic designer uses a transformation to scale a company logo. The transformation $T$ is represented by the matrix $M = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$.

The original logo has an area of $8 \text{ cm}^2$.

Find the area of the scaled logo.

Under the transformation $T$, a specific point on the scaled logo has coordinates $(5t + 1, 2t - 3)$, where $t \in \mathbb{R}$.

Find, in terms of $t$, the coordinates of the original point on the logo.

A new streaming service, CineStream (C), enters the market, competing with the established service, FilmFlick (F).

Market research shows that each month:

• 10% of CineStream subscribers switch to FilmFlick.

• 30% of FilmFlick subscribers switch to CineStream.

The transition matrix $T$ describing these changes, where the rows and columns are ordered (CineStream, FilmFlick), is given as $T = \begin{pmatrix} 0.90 & 0.30 \\ 0.10 & 0.70 \end{pmatrix}$.

The two eigenvalues for this matrix are $1$ and $0.60$. An eigenvector corresponding to the eigenvalue of $1$ is $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.

Find an eigenvector corresponding to the eigenvalue of $0.60$.

A diagonal matrix of eigenvalues is $D = \begin{pmatrix} 0.60 & 0 \\ 0 & 1 \end{pmatrix}$.

Write down an expression for $D^n$, giving your answer as a $2 \times 2$ matrix in terms of $n$.

When CineStream and FilmFlick first launched, there were a total of $8000$ users, all of whom initially subscribed to CineStream.

Assuming the total number of users remains constant, find an expression for the number of users who will favour FilmFlick after $n$ months.

A digital artist is manipulating an image on a screen. The image undergoes two successive linear transformations. The first transformation, $T_1$, is represented by the matrix $A = \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix}$. The second transformation, $T_2$, is represented by the matrix $B = \begin{pmatrix} 4 & 1 \\ -2 & 0 \end{pmatrix}$.

(i) Determine the single matrix, $M$, that represents the combined transformation of applying $T_1$ followed by $T_2$.

(ii) Find the inverse matrix, $M^{-1}$, which would transform the final image back to its original state.

The original image has an area of 49 cm$^2$. Calculate the area of the image after both transformations have been applied.

A scientific experiment involves a platform that oscillates with damped motion. The displacement, $x$, of the platform, measured in centimetres from its equilibrium position, can be modelled by the second order differential equation:

$\ddot{x} + 5\dot{x} + 4x = 0$, where $t$ is the time in seconds after the initial displacement.

Given that $y = \dot{x}$, show that $\dot{y} = -4x - 5y$.

The differential equation can be expressed in the form $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix}$, where A is a $2 \times 2$ matrix.

Write down the matrix A.

Find the eigenvalues of matrix A.

Find the eigenvectors of matrix A.

Given that at $t = 0$, the platform is displaced $6$ cm from equilibrium and its velocity is $3$ cm/s, find an expression for $x$ in terms of $t$.

A market research company studies the customer loyalty between two competing streaming services, StreamSphere (S) and CineFlow (C), over a number of months.

Each month, customers either maintain their current subscription or switch to the other service.

In any given month, it is observed that the probability of a StreamSphere customer remaining with StreamSphere the following month is 0.85, and the probability of a CineFlow customer switching to StreamSphere the following month is 0.20.

This situation can be represented by the transition matrix

$\mathbf{T} = \begin{pmatrix} 0.85 & 0.20 \\ 0.15 & 0.80 \end{pmatrix}$

Interpret the value 0.15 in $\mathbf{T}$ in terms of the changes in customer loyalty between the streaming services.

Find the eigenvalues of matrix $\mathbf{T}$.

One of the eigenvectors of $\mathbf{T}$ is $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$.

Find another, non-parallel, eigenvector and interpret it in context.

The concentration of a chemical, $C$ (in mol dm$^{-3}$), in a reaction vessel at time $t$ seconds is modelled by the differential equation

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 0$

(a) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to show that this equation can be written as

$\begin{pmatrix} \frac{\mathrm{d}C}{\mathrm{d}t} \\ \frac{\mathrm{d}V}{\mathrm{d}t} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix} \begin{pmatrix} C \\ V \end{pmatrix}$.

(ii) Find the eigenvalues for the matrix $\begin{pmatrix} 0 & 1 \\ -10 & -7 \end{pmatrix}$.

(iii) Hence state the long-term rate of change of the chemical concentration.

The chemical reaction is now subjected to an external input, and the equation for the concentration is amended to

$\frac{\mathrm{d}^2C}{\mathrm{d}t^2} + 7\frac{\mathrm{d}C}{\mathrm{d}t} + 10C = 2t + 5$.

(b) (i) Use the substitution $V = \frac{\mathrm{d}C}{\mathrm{d}t}$ to write the differential equation as a system of coupled, first order differential equations. State the initial conditions given that, at $t = 0$, the concentration of chemical C is $1$ mol dm$^{-3}$ and its rate of change is $0$ mol dm$^{-3}$ s$^{-1}$.

(ii) Use Euler's method with a step length of $0.1$ to find the concentration of the chemical when $t = 1$ s. Give your answer to three significant figures.

(iii) Find the long-term rate of change of the chemical concentration.

A drone's camera gimbal can rotate to adjust its view. The transformation matrix $R(\theta)$ that rotates a point $(x, y)$ on the camera's image plane counter-clockwise about the origin through an angle $\theta$ is given by:

$R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$

The drone needs to perform a total rotation of $2\theta$ for a panoramic shot. Write down the matrix $R(2\theta)$ that represents this single combined rotation.

Alternatively, the drone's gimbal controller performs two consecutive rotations, each through an angle of $\theta$. Calculate the resulting transformation matrix when $R(\theta)$ is applied twice, i.e., $R(\theta) \times R(\theta)$.

By comparing your results from part (a) and part (b), explain how the identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ can be derived.

Using the same comparison as in part (c.i), and the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, show that $\cos(2\theta) = 1 - 2\sin^2(\theta)$.

(a) An architect is designing a modular facade for a building. Each module is based on a fundamental rectangular panel, $P$. The vertices of panel $P$ are $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$.

Show that the area of the panel $P$ is $6$ square units.

(b) The design incorporates 30 elements, each obtained by transforming the panel $P$. These elements form a symmetric pattern, with the y-axis acting as a line of symmetry.

The transformation that produces each of the elements on the right side of the design can be represented by a matrix of the form

$M_k = \begin{pmatrix} (1-\frac{k}{15})\cos(k\times12\degree) & -(1-\frac{k}{15})\sin(k\times12\degree) \\ (1-\frac{k}{15})\sin(k\times12\degree) & (1-\frac{k}{15})\cos(k\times12\degree) \end{pmatrix}$

where $k = 0, 1, 2, ..., 14$.

(i) Find the matrix $M_0$. Give your answer in the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a, b, c, d \in \mathbb{Q}$.

(ii) Hence find the coordinates of the image of the vertex $(3,2)$ after it is transformed by the matrix $M_0$.

(c) The matrix $M_k$ can be expressed as the product of a rotation matrix and an enlargement matrix.

(i) Write down, in terms of $k$, the rotation matrix.

(ii) Write down, in terms of $k$, the enlargement matrix.

(iii) Write down, in terms of $k$, the angle of the rotation.

(iv) Write down, in terms of $k$, the scale factor of the enlargement.

(d) Using your answer to part (c)(iv), or otherwise, find the determinant of the matrix $M_k$ in terms of $k$.

(e) Hence, or otherwise, find the total area of the elements in the whole design. Give your answer to three significant figures.

(f) Each element on the left side of the design can be obtained through a transformation of the panel $P$ by applying the matrix $N_k$, where $k = 0, 1, 2, ..., 14$.

Write down the matrix $N_k$ as a product of two matrices.

A manufacturing company, 'GlobalTech', tracks its daily production and operational parameters using matrices. The following matrices represent various aspects of their operations:

• $P$: Daily production output from Department Alpha (units of Product X, Product Y)

• $Q$: Daily production output from Department Beta (units of Product X, Product Y)

• $D$: Daily market demand fluctuations (changes in demand for Product X, Product Y)

• $T$: Efficiency transformation matrix for resource allocation across three stages (Stage 1, Stage 2, Stage 3)

The matrices are given as:

$P = \begin{pmatrix} 4 & 2 \\ -1 & 3 \end{pmatrix}$, $Q = \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$, $D = \begin{pmatrix} -2 & 1 \\ 3 & -4 \end{pmatrix}$, $T = \begin{pmatrix} 1 & 0 & -1 \\ 2 & -3 & 1 \end{pmatrix}$

Without using the matrix utility on your GDC, calculate the combined adjusted production output if Department Alpha's output is tripled and Department Beta's output is doubled, represented by $3P + 2Q$.

Without using the matrix utility on your GDC, determine the net difference in production between Department Alpha and Department Beta, represented by $P - Q$.

Without using the matrix utility on your GDC, find the impact of Department Alpha's production on market demand fluctuations, represented by $P D$.

Without using the matrix utility on your GDC, calculate the effect of market demand fluctuations on the efficiency transformation across stages, represented by $D T$.

Without using the matrix utility on your GDC, determine the result of Department Alpha's production being applied twice, represented by $P^2$.

Without using the matrix utility on your GDC, find the inverse of Department Alpha's production matrix, $P^{-1}$.

Without using the matrix utility on your GDC, calculate the adjusted production output for Department Beta if a fixed output reduction of 3 units for each product is applied, represented by $Q - 3I$, where $I$ is the $2 \times 2$ identity matrix.

Without using the matrix utility on your GDC, explain why the operation $P T + Q$ is not possible.

(a) A student's study habits are modelled by a Markov chain. If the student studies Mathematics (M) on a given day, the probability they study Mathematics the following day is $0.75$. If the student studies Physics (P) on a given day, the probability they study Physics the following day is $0.85$.

Write down a transition matrix, $T$, that shows the movement of the student's study focus between Mathematics and Physics.

(b) On Monday, a student spent their study time on Mathematics. Find the probability that the student will be studying Physics on Friday.

(c) Write down the characteristic polynomial for the matrix $T$. Give your answer in the form $\lambda^2 + b\lambda + c = 0$.

(d) Calculate the eigenvectors for the matrix $T$.

(e) Write down matrices $P$ and $D$ such that $T= PDP^{-1}$, where $D$ is a diagonal matrix.

(f) Hence, find the long-term probability that the student is studying Physics.

A team of researchers is studying the population dynamics of two interacting species, represented by a system of linear differential equations. The transition matrix for the system over a certain period is given by $A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$.

(a) Show that the matrix $A$ satisfies the equation $A^2 - 7A + 10I = 0$, where $I$ is the $2 \times 2$ identity matrix.

(b) Hence express $A^3$ in the form $pA + qI$.

(c) Hence express $A^4$ in the form $rA + sI$.

A competitive market has two dominant smartphone brands, Aura and Zenith. Each year, it is observed that $15\%$ of Aura customers switch to Zenith, while $5\%$ of Zenith customers switch to Aura. All other customer movements are negligible.

Write down a transition matrix $T$ representing the customer movements between the two brands in a particular year. Assume the order of brands is Aura, then Zenith.

Find the eigenvalues and corresponding eigenvectors of $T$.

Hence write down matrices $P$ and $D$ such that $T = PDP^{-1}$.

Initially, Brand Aura has $100\,000$ customers and Brand Zenith has $200\,000$ customers.

Find an expression for the number of customers Brand Aura has after $n$ years, where $n \in \mathbb{N}$.

Hence write down the number of customers that Brand Aura can expect to have in the long term.

A furniture workshop produces three types of items: chairs, tables, and cabinets. Each item requires a specific amount of three different raw materials: wood, metal, and fabric.

To produce one chair, the workshop needs $2$ units of wood, $1$ unit of metal, and $3$ units of fabric.

To produce one table, it needs $1$ unit of wood, $2$ units of metal, and $1$ unit of fabric.

To produce one cabinet, it needs $3$ units of wood, $1$ unit of metal, and $2$ units of fabric.

In a particular week, the workshop used a total of $49$ units of wood, $28$ units of metal, and $51$ units of fabric.

Let $x$, $y$, and $z$ represent the number of chairs, tables, and cabinets produced, respectively.

(a) Write this system of equations in the form $AX = B$, where $A$, $X$, and $B$ are matrices.

(b) Find the inverse matrix $A^{-1}$.

(c) Hence, determine the number of chairs, tables, and cabinets produced that week.

The following question explores a possible method of drawing phase portraits for non-linear coupled systems, taking a predator-prey model as a particular example.

A forest ecosystem contains a population of rabbits ($x$, measured in hundreds), and a population of wolves ($y$, measured in hundreds).

Research indicates that the population dynamics of both rabbits and wolves can be modelled by the following differential equations, in which $t$ is measured in years.

$\frac{dx}{dt} = 3x - \frac{xy}{2}$

$\frac{dy}{dt} = -2y + \frac{2xy}{5}$

for $x, y \ge 0$

At a specific time, there are 400 rabbits and 400 wolves, represented here by the coordinate pair (4, 4). At this time, determine the rate of change of

rabbits.

wolves.

There are two equilibrium points for the populations: A(0, 0) and B ($p, q$).

Explain why A is an equilibrium point.

Find the value of $p$ and the value of $q$.

At points close to A(0, 0), we can ignore the $xy$ terms, so that the system can be approximated by:

$\frac{dx}{dt} = 3x$

$\frac{dy}{dt} = -2y$

for $x, y \ge 0$.

By solving these two differential equations,

find an expression for $x$ in terms of $t$.

find an expression for $y$ in terms of $t$.

Using your answers from part (c), show that phase portrait trajectories close to A may be given by the equation $x^2y^3 = k$, where $k$ is a positive constant.

Hence sketch, on a phase portrait, one possible trajectory for small values of $x$ and $y$.

Now consider points $(x, y)$ close to B on the phase plane. These coordinates can be rewritten as $x = p + X$ and $y = q + Y$, where $p$ and $q$ are the values from part (b)(ii).

By substituting into the original model, show that, for small values of $X$ and $Y$:

$\dot{X} \approx -\frac{5Y}{2}$

Similarly, it can be shown that $\dot{Y} \approx \frac{12X}{5}$.

Given that $\begin{pmatrix} \dot{X} \\ \dot{Y} \end{pmatrix} = M \begin{pmatrix} X \\ Y \end{pmatrix}$, where $M$ is a square matrix, write down $M$.

By finding the eigenvalues of $M$, describe the path of the trajectories close to point B.

Hence sketch a complete set of trajectories in the phase plane for the original model, clearly indicating both equilibrium points.

In this forest ecosystem, at a specific time, there are 400 rabbits and 400 wolves.

Based on the values found in part (a), the wildlife keeper is worried and assumes that the wolves will quickly die out. Suggest whether this assumption is supported by the model. Justify your answer.

A robotics engineer is programming a swarm of drones. The initial paths of the drones are given by a family of lines, $L_k$, with equation $y = kx + (5-2k)$, where $k$ is a real parameter and $k \neq -1.5$.

A control signal applies a linear transformation to the coordinates of every drone, described by the matrix $T = \begin{pmatrix} 3 & 2 \\ -6 & -4 \end{pmatrix}$.

The new path of a drone is the line $L'_k$.

(a) Find a vector equation for the line $L_k$ in terms of $k$.

(b) Find the determinant of $T$.

(c) Show that all the drones will end up moving along the same path, by showing that the equation of the transformed line $L'_k$ does not depend on $k$.