Matrices and Transformations Questions and Answers

Matrices and Transformations Questions

1.  a) (i) On the grid provided, with the same scale on both axes, draw the square S whose vertices are (0, 0), (2, 0), (2,2) and (0, 2). (1 mk)

(ii) Find the coordinates and draw the image T of S under the   transformation whose matrix A maps a point (x, y) onto (x’, y’)         where;     (3 mks)

(iii) Draw the image U of S under the transformation whose matrix is     (2 mks)

(b) (i) Find the product AB and draw the image V of S under the   transformation whose matrix is AB   (3 mks)

 (ii) Describe the single transformation that maps S onto V   (1 mk)

2.  On the grid provided, draw triangle PQR with P(2,3), Q(1,2) and R(4,1). On the same axes draw  triangle P¹¹Q¹¹R¹¹ with vertices P¹¹(-2,3), Q¹¹(-1,2) and R¹¹(-4,1).  (2mks)

 (a) Describe fully a single transformation which will map triangle PQR onto triangle P¹¹Q¹¹R¹¹. (1mk)

  (b) On the same plane, draw triangle P¹Q¹R¹ the image of triangle PQR under reflection in  the line y = -x. (2mks)

 (c) Describe fully a single transformation which maps triangle P¹Q¹R¹ onto triangle P¹¹Q¹¹R¹¹. (2mks)

(d) Draw triangle P¹¹¹Q¹¹¹R¹¹¹ such that it can be mapped onto triangle PQR by a position quarter about (0,0) (2mks)

 (e) State all pairs of triangles that are oppositely congruent. (1mk)

3.  a) Given the transformation matrices

T1 = 2 1 and T2 = 3 1

-1 -2   1 3

 and that transformation T₁ followed by T₂ can be replaced by a single transformation T, write

 down the matrix for T. (3 marks)

1. Find the inverse of matrix T (2 marks)

1. The points A¹¹(7,-11), B¹¹(-7,-13), C¹¹(-8,16) and D¹¹(8,8) are the images of points A, B, C and D respectively under transformation T₁ followed by T2

   Write down the co-ordinates of A, B, C, and D. (5 marks)

4.  A(3, 7), B(5, 5), C(3, 1), D(1, 5)

1. On the grid provided in the next page, plot ABCD on a Cartesian plane  (2mks)
2. A’B’C’D’ is the image of ABCD under a translation. Plot A’B’C’D’ and state its coordinates. (2mks)
3. Plot A”B”C”D”, the image of A’B’C’D’ after a rotation about (-1, 0) through a positive quarter turn. State its coordinates.  (3mks)
4. A”’B”’C”’D”’ is the image of A”B”C”D” after a reflection in the line y=x + 2. Plot A”’B”’C”’D”’ and state its coordinates (3 mks)

5.  A transformation represented by the matrix maps P(0,0), Q(2,0), R(2,3) and S(0,3) onto P’, Q’, R’, S’

1. On the grid provided draw the quadrilateral PQRS and P’Q’R’S’   (4mks)
2. (i) Determine the area of PQRS (1mk)

   (ii) Hence or otherwise find the area of P’Q’R’S’ (2mks)

   c)  A transformation represented by the matrixmaps P’Q’R’S’ onto P”Q”R”S”. Determine the matrix of transformation that would map P”Q”R”S” onto PQRS  (3mks)

6.  A translator T maps P (8, -2) onto P¹ (-2, -3). Find the image of Q (6, -2) under the same translation. (3 mks)

7.  The vertices of a triangle are A(2,5), B(4,3) and C(2,3). H represents a half turn rotation about the point (0,2).

1. Draw triangle ABC and under H  (4 marks)
2. T represents a reflection in the lone x= 0 and K represent a translation. Find the coordinates of of under TK. Hence draw (4 marks)
3. Describe a single transformation that maps ABC onto (2 marks)

8.  Given triangle ABC with vertices A (-6, 5), B(-4, 1) and C(3, 2) and that A(-6, 5) is mapped

  onto A¹(-6, -4) by a shear with y-axis in variant. On the grid provided below;

   (i) draw triangle ABC

(ii) draw triangle A¹B¹C¹, the image of triangle ABC, under the shear

(iii) determine the matrix representing the shear

 (b) Triangle A¹B¹C¹ is mapped onto A¹¹B¹¹C¹¹ by a transformation defined by the matrix

  (i) Draw triangle A¹¹B¹¹C¹¹ on the same grid as ABC and A¹B¹C¹

  (ii) Describe fully a single transformation that maps A¹¹ B¹¹C¹¹  

9. (a) Under a certain rotation A( 2,0) is mapped onto A¹(-4, 2) and B(0,5) is mapped onto B¹(-9, o)

(i) On the grid provided plot the lines AB and A¹B¹ on the same axes

(ii) Hence determine by construction the co-ordinates of the centre and angle of rotation

(b) Under a quarter positive turn about the origin O, A¹ is mapped onto A¹¹ and B¹ is mapped

onto B¹¹. Determine the co-ordinates of A¹¹ and B¹¹

(c) Describe fully a single transformation which would map A to A¹¹ and B to B¹¹    

10.   A transformation T is represented by the matrix and transformation U by the

matrix. Given that a rectangle has co-ordinates at A (1,2) B(6, 2), C(6, 4) and D (1, 4) and

that under T the image of ABCD is A₁B₁C₁D and under U the image of A₁B₁C₁D is A₂B₂C₂D₂:

 (a) Find the co-ordinates of A₁B₁C₁D₁ and A₂B₂C₂D₂

 (b) On the grid provided, plot ABCD, A₁B₁C₁D₁ and A₂B₂C₂D₂  

 (c) Describe the transformation represented by:-

 (i) U

(ii) UT

 (d) If A₂B₂C₂D₂ were to be transformed by a transformation represented by the matrix

to map onto A₃B₃C₃D₃ . What would be the area of A₃B₃C₃D₃  

11.  The vertices of a quadrilateral are A(2,2) B(8,2) , C (8,6) and D(6,4) under a rotation the

images of vertices A and D are A(0,8) and D1(-2, 12).

 (a) On the grid provided and using the same axes draw the quadrilateral ABCD and the

points A¹ and D¹

 (b) Determine the centre and angle of rotation

 (c)Locate the points B¹ and C¹ under the rotation and complete the quadrilateral

12.  A translation maps the point P(5, -3) onto P¹(2, -5)

(a) Determine the translation vector T (b) A Point R¹ is the image of R(-2, -3) under the same translation in (a) above, find the

magnitude of P¹R¹

13.  Triangle ABC has vertices at A(0, -1), B(4, 3)and C(2,2).

 (a) Find the coordinates of image triangle A¹B¹C¹ of triangle ABC under translation  

 (b) Given that triangle A¹¹B¹¹C¹¹ is the image of triangle A¹B¹C¹ under an enlargement

scale factor 3, centre O(0,0) , find the coordinates of A¹¹, B¹¹and C¹¹

 (c) If the area of triangle A¹B¹C¹ is 24 cm², calculate the area of triangle A¹¹B¹¹C¹¹  

 (d) Find the matrix that maps triangle A¹¹B¹¹C¹¹ onto triangle ABC  

14.  a) The triangle ABC where A (2,-1) B (1, 2) and C (4, 4) is reflected in the line X = 4

to give triangle A₁B₁C₁. Draw the two triangles on the graph provided and state the

co-ordinates of A₁B₁C₁

b) Draw the triangle A₂ (5,6), B₂ (2,7) and C₂ (0,4). Given that triangle A₂B₂C₂ is the image of

triangle A₁B₁C₁ under rotation, determine the centre and angle of this rotation

 c) Show the image of triangle A₂B₂C_2, under an enlargement centre (0, 6) scale factor -1

15.  (a) Find the co-ordinates for the image of point P(6, -2) under the transformation defined by :-

 x¹ = x – 3y

y¹ = 2x  

 (b) (i) A quadrilateral ABCD has vertices A(4, -3), B(2, -3), C(4, -1) and D(5, -4). On the grid

 provided, draw the quadrilateral ABCD

  (ii) A¹B¹C¹D¹ is the image of ABCD under a rotation through +90^o about the origin.

On the same axes, draw A¹B¹C¹D¹under the transformation  

`  (c) A²B²C²D²is the image of under A¹B¹C¹D¹ under another transformation by the matrix

(i) Determine the co-ordinates of A²B²C²D² and plot it on the same axes

(ii) Describe the transformation that maps A¹B¹C¹D¹onto  A²B²C²D²

(d) Find a single matrix of transformation that would map A²B²C²D²onto ABCD

16.  (a) Triangle XYZ has vertices X(2, -1) Y(4, -1) and Z (4,2). Triangle XYZ maps onto triangle

X¹Y¹Z¹ under transformation T₁ = . Draw triangles XYZ and its image X¹Y¹Z¹ on

the grid provided  

 (b) Another triangle X¹¹Y¹¹Z¹¹ is the image of X¹Y¹Z¹ after transformation T_{2 =}
.

  Draw triangle X¹¹Y¹¹Z¹¹on the same set of axes

 (c) Find the single transformation matrix T that maps triangle XYZ on to the final image

X¹¹Y¹¹Z¹¹

 (d) Given that the area of triangle XYZ is 15cm², find the area of the triangle X¹¹Y¹¹Z¹¹  

17.  The quadrilateral A (2,1), B (4,1), C (4,4) and D (2,4) is mapped onto A’B’ C’D’ by a matrix

M₁ such that A¹ (8,7), B¹ (14,7), C¹ (14,16) and D¹ (8,16) .

 a) Draw both ABCD and A¹B¹ C¹D¹ on the same plane

 b) Find the matrix of transformation that mapped ABCD onto A’B’ C’D’ and describe it fully  

 c) A¹B¹ C¹D¹ underwent another matrix transformation at N which is a translation that gave

the image A¹¹ B¹¹ C¹¹ D¹¹, Where A¹¹ (7,9), B¹¹ (13,9), C¹¹ (13,18) and D¹¹ (7,18).

The transformation N is a translation . Find the translation  

 d) Draw A¹¹ B¹¹ C¹¹ D¹¹ on the same axes where ABCD and A¹B¹ C¹D¹ were drawn  

18.  a) On the grid provided. Plot the points A(2, -1) B (0, -3) C(2, -4) and D (4, -2) and join them to

form a quadrilateral ABCD. What is the name of this quadrilateral?

b) The points A¹ (1, 2) B¹ (3, 0) C¹ (4, 2) and D¹ (2, 4) are the images of ABC and D under a

certain transformation T₁. On the same grid draw quadrilateral A¹B¹C¹D¹ and describe

transformation T₁ fully.

 c) The points A¹¹(-2, -4) B¹¹(-6, 0) C¹¹(-8, -4) and D¹¹(-4, -8) are the images of A¹B¹C¹D¹ under

transformation T₂. On the same grid draw quadrilateral A¹¹B¹¹C¹¹D¹¹ and describe the

transformation T₂ fully.

d) On the same grid draw quadrilateral A¹¹¹ B¹¹¹ C¹¹¹ D¹¹¹, the image of A¹¹ B¹¹ C¹¹ D¹¹ under a

reflection in the x-axis. State the co-ordinates of A¹¹¹ B¹¹¹ C¹¹¹ D¹¹¹.

19. The Points A¹B¹ and C¹ are the images of A(4, 1), B( 0, -2) and C( -2, 4) respectively

under a transformation represented by the matrix;

M = -1 1

2 -3

(a) Write down the coordinates of A¹ B¹ and C¹

(b) A¹¹ B¹¹ and C¹¹ are the images of A¹ B¹ and C¹ under another transformation whose

Matrix is:

2 -1

1 2 Write down the coordinates of  A¹¹ B¹¹ and C¹¹

 (c) Transformation M followed by N can be represented by a single transformation P.

  Determine the matrix for P

 (d) A matrix P is given by  8 7

 4 5

Find P ^-1

20.  Triangle A¹B¹C¹ is the image of triangle ABC under a transformation represented by matrix

T = If the area of triangle A¹B¹C¹ is 25.6cm², find the area of the object

21.  A point P(2, -4) is mapped into P¹(4, 0) under a translation.

 Determine the image of point Q(-1, 2) under the same translation  

22.  The points A (2, 6), B (1, 1), C (2, 3) and D (4,0) are the vertices of quadrilateral ABCD.

 (a) On graph paper plot the points A, B, C, and D and join them to form quadrilateral ABCD.

 (b) The points A, B, C and D are the images of A¹, B¹, C¹ and D¹ respectively under an

  enlargement centre the origin and scale factor -2. On the same grid draw the image

quadrilateral A¹ B¹ C¹ D¹.

(c) The points A¹¹ B ¹¹ C¹¹ and D¹¹ are the images of ABCD respectively under reflection in the

x – axis. On the same grid, locate the pints A¹¹ B¹¹ C¹¹ and D¹¹ and draw the second image

quadrilateral A¹¹ B ¹¹ C¹¹ D¹¹.

(d) Quadrilateral A¹¹¹ B¹¹¹ C¹¹¹ D¹¹¹ is the image of ABCD under a certain transformation T.

Describe transformation T fully.  

23.  T is a transformation represented by the matrix . Under T, a square of area

10cm² is mapped onto a square 110cm². Find the values of x

Matrices and Transformations Answers

1.  a) B (4,-5), C (3,6 ½ )

   ∆ ABC drawn

∆ ABC drawn

a) ii) Shear maps

I (1, 1½ )

 Matrix = 1 0

1 ½

b) i) A B C

-1 0 -6 -4 3

³/₂ -1 -4 -5 6 ½

=  A¹¹ B¹¹ C¹¹

6 4 -3

    -5 -1 -2

∆ A¹¹ B¹¹ C¹¹ D¹¹ drawn

ii) Half turn about (0,0)

2.  

(a) Centre (-2, -2) 90^o

(b) A11 (-2 , -4) , B11 (0, 9)

(c) Half-turn about the centre (0, 2)

3.  

(b)

(c) (i) U – – positive three-quarter turn about the origin

  (ii)UT – Reflection I the line x = 0

(d) IdetI = I2.5 x -2 – 1x 0 I= 5

Area = 5x(5×2) = 20sq. units

. (a)

b) Centre (-2, 4)

  Angle + 90°

5.  P(5,-3) P¹ (2, -5)

5 + a = 2

-3 b -5

a = -3

b -2

R¹= -2 + -3

-3 -2

= -5

-5

P¹R¹ = -5 – -2

-5 -5

= -7

0

Mag. = 7units

6.  A¹ = (0+1, -1-2) = (1, -3)

B¹ = (4 + 1) , 3-2) = (4, 1)

C¹ = ( 2 +1, 2-2) = (3-0)

Matrix 3 0

 0 3

3 0   1   5  3 = 3  15   9

0 3  -3   1  0  -9   3   0

A¹¹ (3, -9) B¹¹(15, 3) C¹¹(9,0)

Determinant (0-9) =-9

Area = 9×24 = 216cm²

a b   3  15  =   1  5

c d   -9  3 -3  1

5(31 -9b =1  5(3c-9d=-3

-15a+3b =5  15c +3d=1

-48b =0 -48d = -16

b = 0 d = ¹/₃

a = ¹/₃ c=0

matrix  ¹/₃  0

0  ¹/₃

7.  Scale used S₁

ABC drawn B₁

A₁B₁C₁ drawn B₁

A, (6, -1), B(7, 2) C, (4, 4) B1

Line x = 4 L₁

A₂ B₂ C₂ drawn B₁

Two seen B₁

Centre of rotation

Angle of centre of rotation B₁

A₃B₃C₃ drawn B₁

Scale used S₁

ABC drawn B₁

A₁B₁C₁ drawn B₁

A, (6, -1), B(7, 2) C, (4, 4) B1

Line x = 4 L₁

A₂ B₂ C₂ drawn B₁

Two seen B₁

Centre of rotation

Angle of centre of rotation B₁

A₃B₃C₃ drawn B₁

8.  (a)  P(6, -2)

X¹ = 6 -3 (-2) = 12

Y¹ = 2(6) = 12

(X¹, Y¹) = (12, 12)

(b)  (i) A¹(3, 4)

(ii) B¹ (3, 2)

C¹ (1, 4)

D¹(4, 3)

(c) (i)

=

A¹¹ (-5, 4) , B¹¹(-1, 2), C¹¹(-7, 4) and D¹¹(-6, 5)

(ii) A stretch with y-axis invariant and a sketch factor (3)

2h = 6

h = 3

-5a + 4b = 4 -5c + 4d = -3

– a + 2b = 2
-c + 2d = 3

-5a + 4b = 4 -5c + 4d = -3

-a + 4b = 4
-c + 4d = -6

-4a = 0 – 4c = 3

a = 0 c = – ¾

b = 1 d =¹⁵/₈

9.  (a)  X1(5, -1) y₁(7, -1) Z₁ (-2, 2)

xyz & x₁y₁z₁ well drawn

(b) 1-3 xyz x1y1z1

X₂(2, 10) y₂(2, 14)

X₂y₂Z₂ well drawn

(c)

(d)) Area of X₂y₂Z₂

= 4×15 = 60cm²

10.  a b 2 4 4 2 = 7 14 14 8

c d 1 1 4 4 8 7 16 16

 2a + b = 8

4a + b = 14

-2a = -6

6 + b -= 8

b = 2

6 + b = 8

b=2

2c + d = 7

4c + d = 7

-2c = 0

c = 0

d = 7



– it is an enlargement with scale factor 3 with centre (-1, -2)

(c) 8 + a = 7

  7 b 9

a + 8 = 7 7 + b = 9

a =-1 b = 2

T = -1

2

11.  a)  ABCD drawn B₁

Name – Parallelogram B₁

b)  A¹B¹C¹D¹ drawn B₁

Attempt to joining any two points and bisecting. B₁

Description – Rotation + 90⁰. B₁ or quarter turn about (0,0)

  c) A¹¹B¹¹C¹¹D¹¹ drawn. B₁

  Description – Enlargement centre (0, 0) Scale factor –Z. B₁

d) A¹¹¹B¹¹¹C¹¹¹D¹¹¹ – drawn. B₁

Attempt to reflect. B₁

Coordinates

A¹¹¹ = 9-2, 4) C¹¹¹ = (-8, 4) B₁ All correct

B¹¹¹ = (-6, 0) D¹¹¹ (-4, 8)

12.  – 1 1 4 0 -2

2 -3 1 -2 4

-3 -2 6

 5 6 -16

  A^| (-3, 5)
B^|(-2, 6)
C^|(6, -16)

2 -1 -3 -2 6

1 2 5 6 -6

  A^|| B^|| C^||

 -11 -10 18

  7 10 -6

A¹(-11, 7) B¹(-10, 10) C” (18, -6)

MN

= 2 -1 -1 1

1 2 2 -3

= -4 5

3 -5

p-1 = 1 5 -7

    -12 -4 8

   –⁵/₁₂
⁷/₁₂

¹/₃ –²/₃

13.  Det = 2 – 6

= – 4

A.S.F = 4

25.6 = 4

x

x = 6.4cm²

Area of ABC = 6.4cm²

14.  T + (2) = (4)

-4 0

 T = (4 – 2) = (2)

0 + 4 4

 (2) + (-1) = (1)

  4 2 6

Q (1,6)  

16.  5x² + 6 = ¹¹⁰/₁₀

5x² + 6 = 11

x² = 1

x = 1