Matrix transformations (reflections, horizontal/vertical stretches, enlargements, translations, rotations) + scaling area using determinant: notes and practice questions

Transformation matrices are important to understand how areas change. Assume there is a hexagon with an area of $18\ cm^{2},$ which is transformed by matrix $N = \begin{pmatrix} 3 & - 5 \\ 2 & 1 \end{pmatrix}$.

$a$ Find accordingly the area of the image of the hexagon.

$b$ If the image of a point $A$, due to this transformation, has coordinates (5,2), Find the coordinates of point A(x,y).

If a transformation $S$ maps the vector $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} x' \\ y' \end{pmatrix}$ as follows:

$S:\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 4 & - 7 \\ 3 & 5 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 2 \\ - 3 \end{pmatrix}$

$a$ If the image of a point $A(x,y)$, due to this transformation, has coordinates (0,16), Find the coordinates of point $A(x,y)$.

$b$ If a quadrilateral $M$ undergoes the transformation $S$, what happens to the area of its image?

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 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.

In a system analyzing oscillating signals, the initial state of a signal is given by the complex number $w = 1 - i\sqrt{3}$.

(a) Express $w$ in the form $re^{i\theta}$, where $-\pi < \theta \le \pi$.

After a certain operation, the signal's state changes from $w$ to $w^2$.

(b) (i) Describe fully the single transformation (a composition of an enlargement and a rotation about the origin) that maps the point representing $w$ to the point representing $w^2$ on the Argand diagram. State the scale factor of the enlargement and the angle of rotation.

(b) (ii) This transformation can be represented by a matrix. Find and simplify the $2 \times 2$ matrix that represents this transformation.

The signal is considered stable when its state $w^n$ is real and positive.

(c) Find the smallest positive integer, $n$, for which $w^n$ is real and positive.

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 $d$.

A robotic arm in an automated factory is used to engrave intricate patterns on metal sheets. The arm's movements are controlled by a series of transformations in a 2D plane relative to a fixed origin O and a standard x-y coordinate system.

In each case, the robotic arm moves to a new position represented by the following transformations, performed in the listed order:

• a rotation anticlockwise of $\frac{\pi}{3}$ radians about O

• a reflection in the line $y = x$

• a rotation clockwise of $\frac{\pi}{6}$ radians about O.

Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.

Find a single matrix P that defines a transformation that represents the overall change in position.

Find $P^2$.

Hence state what the value of $P^2$ indicates about the effect of repeating the overall transformation.

A triangular pattern ABC is engraved on a metal sheet. After the robotic arm performs the overall transformation, the pattern becomes A'B'C'. Show that the area of triangle ABC is equal to the area of triangle A'B'C'.

Find a single transformation that is equivalent to the three transformations represented by matrix P.

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 design company uses a transformation, $T$, to manipulate elements of a logo. The transformation is represented by $\boldsymbol{r}' = \boldsymbol{Pr} + \boldsymbol{q}$, where $\boldsymbol{P}$ is a $2 \times 2$ matrix, $\boldsymbol{q}$ is a $2 \times 1$ vector, $\boldsymbol{r}$ is the position vector of a point in the plane and $\boldsymbol{r}'$ is the position vector of its image under $T$.

Three key points of a design element, (0, 0), (2, 0) and (0, 2), are transformed to (1, -2), $\left( \frac{3}{2}, \frac{\sqrt{3}}{2} - 2 \right)$ and $\left( 1 - \frac{\sqrt{3}}{2}, -\frac{3}{2} \right)$ respectively.

(a) (i) By considering the image of (0, 0), find $\boldsymbol{q}$.

(ii) By considering the images of (2, 0) and (0, 2), show that

$\boldsymbol{P} = \begin{pmatrix} \frac{1}{4} & -\frac{\sqrt{3}}{4} \\ \frac{\sqrt{3}}{4} & \frac{1}{4} \end{pmatrix}$.

The matrix $\boldsymbol{P}$ can be written as $\boldsymbol{P} = \boldsymbol{RS}$, where $\boldsymbol{S}$ and $\boldsymbol{R}$ are matrices.

$\boldsymbol{S}$ represents an enlargement with scale factor 0.5, centre (0, 0).

$\boldsymbol{R}$ represents a rotation about (0, 0).

(b) Write down the matrix $\boldsymbol{S}$.

(c) (i) Use $\boldsymbol{P} = \boldsymbol{RS}$ to find the matrix $\boldsymbol{R}$.

(ii) Hence find the angle and direction of the rotation represented by $\boldsymbol{R}$.

The transformation $T$ can also be described by an enlargement with scale factor $\frac{1}{2}$, centre $(a, b)$, followed by a rotation about the same centre $(a, b)$.

(d) (i) Write down an equation satisfied by $\begin{pmatrix} a \\ b \end{pmatrix}$.

(ii) Find the value of $a$ and the value of $b$.

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) 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 robotic arm's movement is controlled by transformation matrices. Matrix $A$ represents a counter-clockwise rotation of $\frac{\pi}{3}$ radians about the origin, and matrix $B$ represents a counter-clockwise rotation of $\frac{\pi}{6}$ radians about the origin.

Determine the matrix that represents the combined transformation $AB$ applied three times, i.e., $(AB)^3$.

Find the smallest positive integer $n$ such that applying the combined transformation $AB$ exactly $n$ times results in the robotic arm being in a position equivalent to a $180^\circ$ rotation from its initial orientation.

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 $k$.

(a) A triangular flag $PQR$ has vertices $P(1, 2)$, $Q(4, 2)$, and $R(1, 5)$. It undergoes a sequence of transformations:

1. A reflection across the $y$-axis.

2. An enlargement by a scale factor of $2$ about the origin.

3. A translation by the vector $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$.

Determine a single transformation of the form $X' = AX + b$ that maps the original flag $PQR$ to its final image $P'Q'R'$.

(b) Determine the coordinates of the vertices $P'$, $Q'$, and $R'$ of the transformed flag.

(c) The area of the transformed flag $P'Q'R'$ is $k \times$ the area of the original flag $PQR$. Determine the value of $k$.

The matrices $P$ and $Q$ are defined by $P = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $Q = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$.

(a) Describe fully the single geometrical transformation represented by $P$.

A quadrilateral $K$ is mapped onto quadrilateral $L$ by the composite transformation represented by the matrix product $PQ$. The coordinates of the vertices of $L$ are (0, 0), (0, 2), (−1, 1) and (−1, 3).

(b) Find the coordinates of the vertices of $K$.

(c) (i) Find the area of quadrilateral $K$.

(c) (ii) Hence, find the area of quadrilateral $L$.

The matrix $M = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ represents a combination of transformations:

A rotation of 90° clockwise about the origin;

Followed by a horizontal stretch with scale factor 2, with the y-axis invariant;

Followed by a transformation represented by matrix $F$.

(d) Find matrix $F$.

A robotic arm is programmed to perform a sequence of movements on a workpiece. The position of a specific point on the workpiece is initially at $P(x, y)$. The arm applies two successive transformations.

(a) The first transformation, $T_1$, is represented by the matrix $M_1 = \begin{pmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{pmatrix}$. Describe the nature of the transformation represented by $M_1$.

(b) The second transformation, $T_2$, is represented by the matrix $M_2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The combined effect of $T_1$ followed by $T_2$ can be represented by a single reflection in the line $y = kx$, where $k > 0$. Find the value of $k$.

(a) A digital image processing algorithm applies two consecutive transformations to a pixel's coordinates. First, a scaling transformation $S$ is applied, represented by the matrix $S = \begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$. Then, a rotation transformation $R$ is applied, represented by the matrix $R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

Determine the single matrix $T$ that represents the combined transformation from the initial pixel coordinates to the final pixel coordinates.

(b) After these two transformations, a specific pixel, initially at coordinates $P(x, y)$, ends up at the final coordinates $P''(-2, 6)$.

Find the original coordinates of the pixel $P$.

A graphic designer is applying a sequence of transformations to a digital image. The image is defined by coordinates on a 2D Cartesian plane, with the origin at the centre of the image. The transformations are applied to the coordinates of each pixel in the following order:

1. A clockwise rotation about the origin by 45°.
2. A reflection in the line $y = -x$.
3. An anticlockwise rotation about the origin by 135°.

(a) Write down the 2x2 matrix that represents each of these three transformations.

(b) Find a single matrix, $M$, that represents the overall transformation.

(c) Find $M^2$.

(d) Hence, describe what happens to the image if the entire sequence of transformations is applied twice.

(e) A triangular selection of the image is defined by vertices P, Q, and R. The transformed vertices are P', Q', and R'. Show that the area of the triangle PQR is the same as the area of the triangle P'Q'R'.

(f) Find a single geometric transformation that is equivalent to the combined transformation represented by matrix $M$.