Matrix Transformations (DP IB Maths: AI HL)

Exam Questions

3 hours30 questions
1a
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2 marks

Write down the matrix that represents a rotation of 225 degree clockwise about the origin.

1b
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2 marks

Write down the matrix that represents a reflection in the y-axis.

1c
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2 marks

Find a single matrix that represents the composite transformation consisting of the transformation in part (a) followed by the transformation in part (b).

1d
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2 marks

Hence find the coordinates of the image of the point left parenthesis 4 comma negative 1 right parenthesis after a rotation of 225 degree clockwise about the origin followed by a reflection in the y-axis.

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2a
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2 marks

The diagram below shows a triangle .

q2-3-6-matrix-transformations-diagrams-medium

Write down the position matrix of the triangle ABC.

2b
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3 marks

The triangle ABC is to be mapped to triangle straight A straight apostrophe straight B straight apostrophe straight C straight apostrophe by a single transformation defined by the transformation matrix open parentheses table row cell negative 1 end cell 0 row 0 cell negative 1 end cell end table close parentheses.

Find the position matrix of the mapped image and draw triangle straight A straight apostrophe straight B straight apostrophe straight C straight apostrophe on the diagram.

2c
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2 marks

Describe fully the transformation that triangle ABC has undergone.

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3a
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2 marks

Points in a plane are subjected to a transformation T that transforms a point left parenthesis x comma space y right parenthesis to the point left parenthesis x to the power of apostrophe comma space y to the power of apostrophe right parenthesis, where T is defined by

 T colon open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row 4 0 row 0 cell 0.5 end cell end table close parentheses open parentheses table row x row y end table close parentheses

Describe in words the transformation T.

3b
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2 marks

Find the matrix T to the power of negative 1 end exponent.

3c
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2 marks

Hence find the coordinates of the point left parenthesis x comma space y right parenthesis if open parentheses x to the power of apostrophe comma space y to the power of apostrophe close parentheses equals left parenthesis 12 comma negative 4 right parenthesis.

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4a
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2 marks

A quadrilateral has vertices left parenthesis 2 comma blank 1 right parenthesis comma blank open parentheses 4 comma blank 4 close parentheses comma blank open parentheses 7 comma blank 1 close parentheses and left parenthesis 9 comma space 4 right parenthesis.

Find the area of the quadrilateral.

4b
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2 marks

The quadrilateral undergoes a transformation represented by the matrix open parentheses table row cell negative 2 end cell 1 row cell negative 4 end cell 3 end table close parentheses.

Find the determinant of the transformation matrix.

4c
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2 marks

Hence find the area of the image.

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5
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5 marks

An object undergoes a vertical stretch with scale factor 2 followed by a reflection in the x-axis. The position matrix of the image is open parentheses table row 2 5 4 row 3 cell negative 2 end cell cell negative 1 end cell end table close parentheses.

Use a matrix method to find the coordinates of the object before the transformation.

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6a
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5 marks

The points straight A left parenthesis 7 comma negative 3 right parenthesis and straight B left parenthesis 2 comma space 6 right parenthesis are transformed to become the points straight A apostrophe left parenthesis 18 comma space 8 right parenthesis and straight B apostrophe left parenthesis 12 comma space 16 right parenthesis respectively.

Find the 2 cross times 2 spacematrix T that represents the linear transformation.

6b
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3 marks

Given that point straight C left parenthesis negative 4 comma 5 right parenthesis is transformed by T squared, find the coordinates of the image point C apostrophe.

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7a
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3 marks

An object is reflected in the line y equals fraction numerator square root of 3 over denominator 3 end fraction x.

Write down the matrix that represents the transformation.

7b
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3 marks

straight P is a vertex of the object that is being reflected.

Find the coordinates of straight P if the coordinates of its image straight P apostrophe are left parenthesis 2 comma blank 2 square root of 3 right parenthesis.

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8a
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2 marks

The triangle PQR, with vertices straight P left parenthesis negative 1 comma space 1 right parenthesis comma space straight Q left parenthesis 5 comma space 3 right parenthesis and straight R left parenthesis negative 9 comma negative 2 right parenthesis comma is translated by the vector open parentheses table row cell negative 7 end cell row 2 end table close parentheses and then enlarged by a scale factor of 3 with the centre of enlargement at the origin.

Find a single transformation in the form A X plus b that maps PQR onto straight P apostrophe straight Q apostrophe straight R apostrophe.

8b
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3 marks

Hence determine the coordinates of straight P apostrophe straight Q apostrophe straight R apostrophe.

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9a
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3 marks

Find the 2 cross times 2 transformation matrices that represent the following transformations:

(i)
R, a rotation of ​ pi over 4 radians anti-clockwise
(ii)
S, a reflection in the line y equals x

(iii)
T, a stretch with scale factor 5 parallel to the y-axis.
9b
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4 marks

Find a single transformation matrix that represents the composite transformation 

(i)
R T cubed 

(ii)
R to the power of 8 S T S
9c
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2 marks

Find the coordinates of the image of the point straight A left parenthesis 3 comma negative 1 right parenthesis after it has undergone the composite transformation specified in part (b)(i).

9d
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1 mark

State the name of the single transformation that is equivalent to the composite transformation specified in part (b)(ii).

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10a
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3 marks

The triangle PQR with position matrix T subscript 0 has vertices straight P left parenthesis 5 comma space 2 right parenthesis comma straight Q left parenthesis negative 3 comma space 1 right parenthesis and straight R left parenthesis negative 5 comma negative 4 right parenthesis.  

The triangle is transformed by the transformation matrix  M equals open parentheses table row 0 cell 1 half end cell row cell negative 1 half end cell 0 end table close parentheses.  

 T subscript n denotes the position matrix of the image triangle after PQR has been transformed  times by matrix M.

By multiplying two appropriate single transformation matrices together, verify that the matrix M is an enlargement by scale factor 1 half followed by a 90 degree clockwise rotation.

10b
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2 marks

Explain why the area of the triangle with position matrix T subscript 1 will be 1 fourth of the area of triangle PQR.   

10c
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3 marks

Find T subscript 2 comma  and hence the coordinates of the image triangle after triangle PQR is transformed twice by matrix M.

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1a
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3 marks

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

1b
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2 marks

Hence find the coordinates of the image of the point open parentheses negative 7 comma 5 close parentheses, which undergoes the composite transformation stated in part (a).

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2a
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3 marks

The diagram below shows the triangle ABC.

mi_q2a_3-6_matrix-transformations_hard_ib_ai_hl_maths_dig

Write down the matrix that will rotate the triangle about the origin so that open square brackets AC close square brackets is parallel to the x-axis.

2b
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3 marks

Find the position matrix of the mapped image.

2c
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2 marks

Hence find the area of the triangle.

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3a
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2 marks

Points in a plane are subjected to a transformation T colon open parentheses table row x row y end table close parentheses rightwards arrow from bar open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses comma where T is defined by:

 T colon open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell 1 half end cell cell fraction numerator square root of 3 over denominator 2 end fraction end cell row cell fraction numerator square root of 3 over denominator 2 end fraction end cell cell negative 1 half end cell end table close parentheses open parentheses table row x row y end table close parentheses plus open parentheses table row p row 6 end table close parentheses           

Given that a point A open parentheses 4 comma space q close parentheses is mapped to A to the power of apostrophe open parentheses 7 minus square root of 3 comma space 2 square root of 3 plus 7 close parentheses , find p and q, where p comma q element of straight real numbers.

3b
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4 marks

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

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4a
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3 marks

A parallelogram with a base length of 2.5 cm undergoes a transformation represented by the matrix open parentheses table row cell negative 8 end cell 3 row 7 cell negative 3 end cell end table close parentheses 

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

4b
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2 marks

Hence find the perpendicular height of the original parallelogram.

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5a
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3 marks

A function f  is defined by f open parentheses x close parentheses equals x squared.

Given that g open parentheses x close parentheses equals 1 half f open parentheses 4 x close parentheses , describe fully the transformation that maps f open parentheses x close parentheses rightwards arrow from bar g open parentheses x close parentheses.

5b
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3 marks

Hence find the single matrix that represents this transformation.

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6
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6 marks

The points A(4, 9) and B(-3, -11) are transformed by T  to become the points A apostrophe open parentheses 3 comma 8 close parentheses and B apostrophe open parentheses 2 comma negative 6 close parentheses respectively.

Given that the point C open parentheses negative 5 comma negative 7 close parentheses is transformed by T cubed, find the coordinates of the image point C '.

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7a
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5 marks

Consider matrix A, where A equals open parentheses table row 4 2 row cell negative 3 end cell 6 end table close parentheses,  which represents a series of transformations in the following order: 

        • A transformation represented by the matrix open parentheses table row 4 2 row 3 cell negative 6 end cell end table close parentheses
        • A counter-clockwise rotation of 270 degree
        • A single transformation represented by the matrix B 

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

7b
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2 marks

Find B to the power of negative 1 end exponent.

7c
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2 marks

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

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8a
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1 mark

The triangle PQR with vertices straight P open parentheses 4 comma negative 1 close parentheses, straight Q open parentheses 2 comma space 6 close parentheses and straight R open parentheses negative 4 comma space 2 close parentheses  is enlarged by a scale factor of 5 with the centre of enlargement at straight X open parentheses negative 3 comma negative 1 close parentheses. The enlarged shape has coordinates straight P apostrophe straight Q apostrophe straight R apostrophe.

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

8b
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2 marks

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

8c
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2 marks

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

8d
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2 marks

Hence determine the coordinates of straight P apostrophe straight Q apostrophe straight R apostrophe.

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9a
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4 marks

The matrices R comma space S and T  are defined by R equals open parentheses table row cell begin inline style 1 half end style end cell cell begin inline style fraction numerator square root of 3 over denominator 2 end fraction end style end cell row cell negative begin inline style fraction numerator square root of 3 over denominator 2 end fraction end style end cell cell begin inline style 1 half end style end cell end table close parentheses, S equals open parentheses table row 2 0 row 0 3 end table close parentheses and T equals open parentheses table row 1 0 row 0 cell 4 over 9 end cell end table close parentheses.  

Triangle X is mapped onto triangle Y  by the transformation represented by R cubed S T S.

Describe in full the two geometric transformations, A and B, that are equivalent to R cubed S T Sand map triangle X onto triangle Y .

 

9b
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3 marks

The coordinates of triangle Y are open parentheses 2 comma negative 8 close parentheses comma space open parentheses 2 comma space 12 close parentheses and open parentheses 14 comma space 3 close parentheses.

Find the area of triangle X.

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10a
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3 marks

The quadrilateral PQRS with position matrix T subscript 0 has vertices straight P open parentheses 0 comma 0 close parentheses comma space space straight Q open parentheses 0 comma 9 close parentheses comma space straight R open parentheses 6 comma 9 close parentheses, and S open parentheses 6 comma 0 close parentheses.

T subscript n denotes the position matrix of the image quadrilateral after PQRS has been transformed n times by matrix M.

After a transformation, represented by the matrix M, where M equals A X plus b, the position matrix of the image quadrilateral T subscript 1has vertices straight P apostrophe open parentheses 6 comma negative 1 close parentheses comma space straight Q apostrophe open parentheses 6 comma space 5 close parentheses comma space straight R apostrophe open parentheses 10 comma space 5 close parentheses and S apostrophe open parentheses 10 comma negative 1 close parentheses.

Find M . 

10b
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2 marks

Find the position matrix of the image quadrilateral T subscript 2.

10c
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3 marks

Find the perimeter of the shape formed by the quadrilaterals T subscript 0 comma space T subscript 1and  T subscript 2.

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1a
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3 marks

A geometric transformation T colon open parentheses x space y close parentheses rightwards arrow from bar open parentheses x apostrophe y apostrophe close parentheses is defined by

T colon open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell fraction numerator 1 over denominator square root of 2 end fraction end cell cell negative fraction numerator 1 over denominator square root of 2 end fraction end cell row cell negative fraction numerator 1 over denominator square root of 2 end fraction end cell cell negative fraction numerator 1 over denominator square root of 2 end fraction end cell end table close parentheses open parentheses table row x row y end table close parentheses

Given that T is a composite function comprising a transformation defined by the matrix A followed by a rotation of  begin inline style straight pi over 4 end style rad clockwise:

(i)
Find A
(ii)
Describe fully the single geometric transformation represented by A.
1b
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4 marks

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 equals x.

Find B.

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2
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8 marks

A trapezoid A B C D is shown in the diagram below.mi_q2_3-6_matrix-transformations_vhard_ib_ai_hl_maths_dig

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

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3
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7 marks

Points in a plane are subject to a transformation A B that transforms a point open parentheses x comma y close parentheses to the point open parentheses x apostrophe comma space y apostrophe close parentheses, where A and Bare defined by

 A equals open parentheses table row 4 6 row 1 cell negative 3 end cell end table close parentheses comma space B equals open parentheses table row cell negative 2 end cell 0 row 0 cell negative 2 end cell end table close parentheses 

The position matrix of a series of transformed points is X apostrophe equals open parentheses table row 24 0 12 row 0 18 cell negative 9 end cell end table close parentheses. 

Given that the final position matrix of the points X apostrophe apostrophe is a reflection in the y-axis from their original position, find X apostrophe apostrophe.

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4
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6 marks

A triangle A B C undergoes a transformation represented by the matrix open parentheses table row cell negative 3 end cell cell negative 1 end cell row 3 5 end table close parentheses after which it has an area of 48 cm2. 

The original triangle A B C 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 A B C after being transformed by A is 84 cm2, find p.

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5
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5 marks

Consider the functions y equals f open parentheses x close parentheses and y equals g open parentheses x close parentheses defined by f open parentheses x close parentheses equals 2 x squared and g open parentheses x close parentheses equals 3 x plus 5. 

Let the transformation of a point be represented by T colon open parentheses table row x row cell f open parentheses x close parentheses end cell end table close parentheses rightwards arrow from bar open parentheses table row x row cell g ring operator f open parentheses x close parentheses end cell end table close parentheses. 

Use a matrix method to determine the coordinates of a point P after undergoing the transformation T, given that P apostrophe open parentheses negative 2 comma space 29 close parentheses.

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6
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6 marks

T is a 2 cross times 2 matrix open parentheses table row 2 b row c d end table close parentheses that represents the transformation of points A open parentheses negative 1 comma 6 close parentheses and B open parentheses 4 comma 2 close parentheses to A apostrophe open parentheses 2 p comma negative 18 close parentheses points and B apostrophe open parentheses 10 comma negative 3 p close parentheses respectively.

Given that point C open parentheses negative 2 comma 5 close parentheses is transformed by T squared, find the coordinates of the image point C apostrophe.

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7
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4 marks

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

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8a
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2 marks

Consider the general complex number z equals x plus y i with position vector open parentheses table row x row y end table close parentheses and a second complex number z subscript 1.

State the two transformations that occur when z is multiplied by z subscript 1.

8b
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2 marks

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

8c
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4 marks

Given that z equals 1 plus 2 i and z subscript 1 equals 3 square root of 3 minus 3 i and using the result from part (b), find the position vector of the result of z z subscript 1.

8d
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2 marks

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

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9
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5 marks

The matrices R comma space S and T are defined by R, a stretch with scale factor 4 and y-axis invariant, S, a reflection in the line y equals negative x and T equals open parentheses table row a b row c d end table close parentheses. 

Triangle X is mapped onto triangle Y by the transformation represented by R cubed T S.

Given triangle Y is a rotation of triangle X by  straight pi over 4 rad clockwise about the origin, find T.

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10a
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6 marks

The triangle P Q R with position matrix T subscript 0 has vertices P open parentheses 0 comma space 4 close parentheses comma space Q open parentheses 8 comma negative 3 close parentheses,  and R open parentheses 7 comma 7 close parentheses. 

The triangle is transformed by a matrix M comprising a counter-clockwise rotation of  straight pi over 3  about the point open parentheses negative 2 comma 1 close parentheses.

Given that T subscript ndenotes the position matrix of the image triangle after P Q R has been transformed n times by matrix M, find T subscript 3.

10b
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4 marks

The same original triangle P Q R is transformed by a matrix N comprising a clockwise rotation of  straight pi over 3  about the origin followed by a translation of open parentheses table row 3 row 4 end table close parentheses. 

R subscript ndenotes the position matrix of the image triangle after P Q R has been transformed n times by matrix N.

Find the distance between the vertices corresponding to the initial point P spaceof the triangle with position matrix T subscript 3 and the triangle with position matrix R subscript 1.

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