AHL 3.9—Matrix transformations Questionbank

Practice AHL 3.9—Matrix transformations with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider three points $P(1, 2, -1)$ , $Q(4, 3, 2)$ , and $R(0, -1, 5)$ . Use vector operations to answer the following questions.

Find the vector $\overrightarrow{PQ}$ .

[2]

Calculate the magnitude of $\overrightarrow{PQ}$ .

[2]

Determine the dot product of vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ .

[3]

Find the cosine of the angle between $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ .

[3]

Verify if the vectors are perpendicular.

[2]

Question 2

HLPaper 2

A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin $O$ and a set of $x$ - $y$ -axes. In each case, the drone moves to a new position represented by the following transformations:

a rotation anticlockwise of $\frac{\pi}{6}$ radians about $O$
a reflection in the line $y = \frac{x}{\sqrt{3}}$
a rotation clockwise of $\frac{\pi}{3}$ radians about $O$ . All the movements are performed in the listed order.

Write down the matrix for each of the three transformations:

(i) rotation anticlockwise of $\frac{\pi}{6}$ radians about $O$

(ii) reflection in the line $y = \frac{x}{\sqrt{3}}$

(iii) rotation clockwise of $\frac{\pi}{3}$ radians about $O$

[3]

Find a single matrix $\boldsymbol{P}$ that defines a transformation that represents the overall change in position by multiplying the three matrices in the correct order.

[2]

Find $\boldsymbol{P}^2$ .

[1]

Hence state what the value of $\boldsymbol{P}^2$ indicates for the possible movement of the drone.

[1]

Three drones are initially positioned at the points $A$ , $B$ and $C$ . After performing the movements listed above, the drones are positioned at points $A'$ , $B'$ and $C'$ respectively.

Show that the area of triangle $ABC$ is equal to the area of triangle $A'B'C'$ .

[3]

Find a single transformation that is equivalent to the three transformations represented by matrix $\boldsymbol{P}$ .

[1]

Question 3

HLPaper 2

Consider a vector line in 2D represented by the vector equation $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ , where $\mathbf{a}$ is a position vector and $\mathbf{b}$ is a direction vector.

Given $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ , find the coordinates of the point on the line when $t = 2$ .

[2]

Determine the equation of the line in the form $y = mx + c$ .

[3]

Question 4

HLPaper 2

An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable $x$ measures the concentration of mercury in micrograms per litre ( $\mu\text{g L}^{-1}$ ) at time $t$ days.

The initial mercury concentration is $0 \, \mu\text{g L}^{-1}$ and the rate of increase is initially $1 \, \mu\text{g L}^{-1} \text{day}^{-1}$ .

Show that the system of coupled first order equations: $\frac{dx}{dt} = y, \quad \frac{dy}{dt} = -2x - 3y$ can be written as the second order differential equation: $\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 2x = 0$

[2]

Find the eigenvalues of the system of coupled first order equations given in part 1.

[3]

Hence find the exact solution of the second order differential equation.

[3]

Sketch the graph of $x$ against $t$ , labelling the maximum point of the graph with its coordinates.

[2]

The river authority decides that fishing should be stopped when the mercury concentration exceeds $0.1 \, \mu\text{g L}^{-1}$ .

Use the model to calculate the total amount of time when fishing should be stopped.

[3]

Write down one reason, with reference to the context, to support this decision.

[1]

Question 5

HLPaper 2

Consider a 2D transformation matrix $A$ that represents a geometric transformation in the plane.

Given the matrix $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ , determine the type of geometric transformation it represents.

[2]

Find the image of the point $(2, 3)$ under the transformation represented by the matrix $A$ .

[3]

Question 6

HLPaper 2

Let $A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}$.

Let $X = B - A^{-1}$ and $Y = B^{-1} - A$.

It is given that $$A^n = \begin{pmatrix} 1 & n & \dfrac{n(n + 1)}{2} \\ 0 & 1 & n \\ 0 & 0 & 1 \end{pmatrix}$$ for $n \in \mathbb{Z}^+$.

The inverse of $A^n$ is given by $$(A^n)^{-1} = \begin{pmatrix} 1 & x & y \\ 0 & 1 & x \\ 0 & 0 & 1 \end{pmatrix}$$ for $n \in \mathbb{Z}^+$.

Find $X$ and $Y$.

[2]

Does $X^{-1} + Y^{-1}$ have an inverse? Justify your conclusion.

[3]

Find $x$ and $y$ in terms of $n$.

[5]

Hence find an expression for $A^n + (A^n)^{-1}$.

[1]

Question 7

HLPaper 2

Consider the transformation matrix $B$ that represents a reflection across the line $y = x$ .

Write down the matrix $B$ that represents this reflection.

[1]

Verify that the matrix $B$ is its own inverse.

[3]

Question 8

HLPaper 1

Consider two vectors:

\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} \quad \text{and} \quad \mathbf{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}

Questions 1, 2 and 3 refer to these vectors.

Find the sum of the vectors $\mathbf{a} + \mathbf{b}$ .

[2]

Determine the magnitude of the vector $\mathbf{a}$ .

[2]

Express the vector ${2\mathbf{a} - 3\mathbf{b}}$ in component form.

[3]

Consider the vectors $\mathbf{u} = \begin{pmatrix} 2 \\ -4 \\ 6 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix}$ .

Check if the vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel. Justify your answer.

[3]

Question 9

HLPaper 1

A geometric transformation $T:\binom{x}{y} \mapsto\binom{x^{\prime}}{y^{\prime}}$ is defined by $T:\binom{x^{\prime}}{y^{\prime}}=\left(\begin{array}{cc} 7 & -10 \\ 2 & -3 \end{array}\right)\binom{x}{y}+\binom{-5}{4}$

Find the coordinates of the image of the point (6,-2).

[2]

Given that $T:\binom{p}{q} \mapsto 2\binom{p}{q}$ , find the value of p and the value of q.

[3]

A triangle L with vertices lying on the xy plane is transformed by T. Explain why both L and its image will have exactly the same area.

[2]

Question 10

HLPaper 2

Consider the iterative process of generating a fractal using the transformation matrices $D_1$ and $D_2$ .

Let $D_1 = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix}$ and $D_2 = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix} + \begin{pmatrix} 0.5 \\ 0 \end{pmatrix}$ . Describe the effect of applying $D_1$ and $D_2$ iteratively on a point in the plane.

[3]