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AHL 3.9—Matrix transformations - IB Questionbank | RevisionDojo

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 π/6 radians about O
a reflection in the line y = x/√3
a rotation clockwise of π/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 π/6 radians about O

(ii) reflection in the line y = x/√3

(iii) rotation clockwise of π/3 radians about O

[3]

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

[2]

Find P².

[1]

Hence state what the value of P² 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 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.

Show that the system of coupled first order equations:

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

can be written as the given second order differential equation.

[2]

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

[3]

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

[1]

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

[2]

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 = $\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{array}} \right)$ andB= $\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 1&1&0 \\ 1&1&1 \end{array}} \right)$ .

Given that X = B – A^–1and Y = B^–1– A,

You are told that ${A^n} = \left( {\begin{array}{*{20}{c}} 1&n&{\frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&n \\ 0&0&1 \end{array}} \right)$ , for $n \in {\mathbb{Z}^ + }$ .

Given that ${\left( {{A^n}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&x&y \\ 0&1&x \\ 0&0&1 \end{array}} \right)$ , for $n \in {\mathbb{Z}^ + }$ ,

find X and Y.

[2]

does X^–1+ Y^–1have an inverse? Justify your conclusion.

[3]

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

[5]

and hence find an expression for ${A^n} + {\left( {{A^n}} \right)^{ - 1}}$ .

[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}

Answer the following questions based on 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]

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

$\mathbf{a} = \begin{pmatrix} 2 \\ -4 \\ 6 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix}$

[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]

AHL 3.9—Matrix transformations Questionbank