Matrix Transformations (DP IB Applications & Interpretation (AI): HL): Exam Questions

Find a single matrix to represent the composite transformation comprising a reflection in the $x$-axis followed by a rotation of  $60°$ counter-clockwise about the origin.

Hence find the coordinates of the image of the point $\left(-7,5\right)$, which undergoes the composite transformation stated in part (a).

The diagram below shows the triangle ABC.

Write down the matrix that will rotate the triangle about the origin so that $\left[\mathrm{AC}\right]$ is parallel to the $x$-axis.

Find the position matrix of the mapped image.

Hence find the area of the triangle.

Points in a plane are subjected to a transformation $T:\left(\begin{array}{c}x\\ y\end{array}\right)\mapsto \left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right),$ where $T$ is defined by:

 $T:\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}p\\ 6\end{array}\right)$           

Given that a point $A\left(4,q\right)$ is mapped to $A^{\text{'}}\left(7-\sqrt{3},2\sqrt{3}+7\right)$ , find $p$ and $q$, where $p,q\in \mathrm{ℝ}$.

Given that $T$comprises two individual transformations describe in full the composite transformation $T$.

A parallelogram with a base length of 2.5 cm undergoes a transformation represented by the matrix $\left(\begin{array}{cc}-8& 3\\ 7& -3\end{array}\right)$ 

Given that the area of the image after the transformation is 48 cm², find the area of the original parallelogram.

Hence find the perpendicular height of the original parallelogram.

A function $f$  is defined by $f\left(x\right)=x^2$.

Given that $g\left(x\right)=\frac{1}{2}f\left(4x\right)$ , describe fully the transformation that maps $f\left(x\right)\mapsto g\left(x\right)$.

Hence find the single matrix that represents this transformation.

The points A(4, 9) and B(-3, -11) are transformed by $T$  to become the points $A\text{'}\left(3,8\right)$ and $B\text{'}\left(2,-6\right)$ respectively.

Given that the point $C\left(-5,-7\right)$ is transformed by $T^3$, find the coordinates of the image point C '.

Consider matrix A, where $A=\left(\begin{array}{cc}4& 2\\ -3& 6\end{array}\right)$,  which represents a series of transformations in the following order: 

• A transformation represented by the matrix $\left(\begin{array}{cc}4& 2\\ 3& -6\end{array}\right)$

• A counter-clockwise rotation of $270°$

• A single transformation represented by the matrix B 

Find matrix B and describe the effect of the transformation it represents in full.

Find $B^{-1}$.

Compare the matrix $B$ with its inverse and explain any similarities that can be observed.

The triangle $\mathrm{PQR}$ with vertices $\mathrm{P}\left(4,-1\right)$, $\mathrm{Q}\left(2,6\right)$ and $\mathrm{R}\left(-4,2\right)$  is enlarged by a scale factor of 5 with the centre of enlargement at $\mathrm{X}\left(-3,-1\right)$. The enlarged shape has coordinates $\mathrm{P}\text{'}\mathrm{Q}\text{'}\mathrm{R}\text{'}$.

Write down the column vector required to translate the centre of enlargement to the origin.

Write down the position matrix of the vertices of the triangle after undergoing the translation stated in part (a).

Find a single transformation in the form $AX+b$ that will map the coordinates of the vertices of the triangle after they have been translated to their final position P'Q'R'.

Hence determine the coordinates of $\mathrm{P}\text{'}\mathrm{Q}\text{'}\mathrm{R}\text{'}$.

The matrices $R,S$ and $T$  are defined by $R=\left(\begin{array}{cc}\frac{1}{2}& \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2}& \frac{1}{2}\end{array}\right)$, $S=\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$ and $T=\left(\begin{array}{cc}1& 0\\ 0& \frac{4}{9}\end{array}\right)$.  

Triangle X is mapped onto triangle Y  by the transformation represented by $R^{3}STS$.

Describe in full the two geometric transformations, $A$ and $B$, that are equivalent to $R^{3}STS$and map triangle $X$ onto triangle $Y$ .

The coordinates of triangle $Y$ are $\left(2,-8\right),\left(2,12\right)$ and $\left(14,3\right)$.

Find the area of triangle X.

The quadrilateral PQRS with position matrix $T_0$ has vertices $\mathrm{P}\left(0,0\right),\mathrm{Q}\left(0,9\right),\mathrm{R}\left(6,9\right)$, and $S\left(6,0\right).$

$T_n$ denotes the position matrix of the image quadrilateral after $\mathrm{PQRS}$ has been transformed n times by matrix M.

After a transformation, represented by the matrix M, where $M=AX+b$, the position matrix of the image quadrilateral $T_1$has vertices $\mathrm{P}\text{'}\left(6,-1\right),\mathrm{Q}\text{'}\left(6,5\right),\mathrm{R}\text{'}\left(10,5\right)$ and $S\text{'}\left(10,-1\right).$

Find M . 

Find the position matrix of the image quadrilateral $T_2$.

Find the perimeter of the shape formed by the quadrilaterals $T_0,T_1$and  $T_2$.

A geometric transformation $T:\left(xy\right)\mapsto \left(x\text{'}y\text{'}\right)$ is defined by

$T:\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

Given that T is a composite function comprising a transformation defined by the matrix A followed by a rotation of  $\frac{\mathrm{\pi }}{4}$ rad clockwise:

(i) Find A. 

(ii) Describe fully the single geometric transformation represented by A.

After being transformed by $T$, an additional transformation $B$ is undergone. The final position of the points is a reflection of their initial position in the line $y=x$.

Find $B$.

A trapezoid $ABCD$ is shown in the diagram below.

By first transforming the trapezoid so that the base is parallel to the y-axis, calculate the area of the trapezoid.

Points in a plane are subject to a transformation $AB$ that transforms a point $\left(x,y\right)$ to the point $\left(x\text{'},y\text{'}\right)$, where $A$ and $B$are defined by

 $A=\left(\begin{array}{cc}4& 6\\ 1& -3\end{array}\right),B=\left(\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right)$ 

The position matrix of a series of transformed points is $X\text{'}=\left(\begin{array}{ccc}24& 0& 12\\ 0& 18& -9\end{array}\right).$ 

Given that the final position matrix of the points $X\text{'}\text{'}$ is a reflection in the y-axis from their original position, find $X\text{'}\text{'}.$

A triangle $ABC$ undergoes a transformation represented by the matrix $\left(\begin{array}{cc}-3& -1\\ 3& 5\end{array}\right)$ after which it has an area of 48 cm². 

The original triangle $ABC$ is then transformed by $A$, where $A$ is defined as a stretch with scale factor 3 parallel to the x-axis and a stretch with scale factor $p$ parallel to the y-axis. 

Given that the area of triangle $ABC$ after being transformed by $A$ is 84 cm², find $p$.

Consider the functions $y=f\left(x\right)$ and $y=g\left(x\right)$ defined by $f\left(x\right)=2x^2$ and $g\left(x\right)=3x+5$. 

Let the transformation of a point be represented by $T:\left(\begin{array}{c}x\\ f\left(x\right)\end{array}\right)\mapsto \left(\begin{array}{c}x\\ g\circ f\left(x\right)\end{array}\right)$. 

Use a matrix method to determine the coordinates of a point $P$ after undergoing the transformation $T$, given that $P\text{'}\left(-2,29\right)$.

$T$ is a $2\times 2$ matrix $\left(\begin{array}{cc}2& b\\ c& d\end{array}\right)$ that represents the transformation of points $A\left(-1,6\right)$ and $B\left(4,2\right)$ to $A\text{'}\left(2p,-18\right)$ points and $B\text{'}\left(10,-3p\right)$ respectively.

Given that point $C\left(-2,5\right)$ is transformed by $T^2$, find the coordinates of the image point $C\text{'}$.

Show that a transformation matrix representing a reflection is a self-inverting matrix

Consider the general complex number $z=x+yi$ with position vector $\left(\begin{array}{c}x\\ y\end{array}\right)$ and a second complex number $z_1$.

State the two transformations that occur when $z$ is multiplied by $z_1$.

Hence write down the single matrix,$T$ that represents the two transformations from part (a).

Given that $z=1+2i$ and $z_1=3\sqrt{3}-3i$ and using the result from part (b), find the position vector of the result of $z{z}_1$.

By multiplying the two numbers together in their complex form, verify that your answer to part (c) is correct.

The matrices $R,S$ and $T$ are defined by $R$, a stretch with scale factor 4 and y-axis invariant, $S$, a reflection in the line $y=-x$ and $T=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. 

Triangle $X$ is mapped onto triangle $Y$ by the transformation represented by $R^{3}TS$.

Given triangle $Y$ is a rotation of triangle $X$ by  $\frac{\mathrm{\pi }}{4}$ rad clockwise about the origin, find $T$.

The triangle $PQR$ with position matrix $T_0$ has vertices $P\left(0,4\right),Q\left(8,-3\right)$,  and $R\left(7,7\right).$ 

The triangle is transformed by a matrix M comprising a counter-clockwise rotation of  $\frac{\mathrm{\pi }}{3}$  about the point $\left(-2,1\right)$.

Given that $T_n$denotes the position matrix of the image triangle after $PQR$ has been transformed $n$ times by matrix $M$, find $T_3$.

The same original triangle $PQR$ is transformed by a matrix $N$ comprising a clockwise rotation of  $\frac{\mathrm{\pi }}{3}$  about the origin followed by a translation of $\left(\begin{array}{c}3\\ 4\end{array}\right)$. 

$R_n$denotes the position matrix of the image triangle after $PQR$ has been transformed $n$ times by matrix $N$.

Find the distance between the vertices corresponding to the initial point $P$of the triangle with position matrix $T_3$ and the triangle with position matrix $R_1$.