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IG.M5 Matrices and Matrix Transformations

EDEXCEL 4MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Questions 1 to 12 do not require full sentence answers. Questions 13 to 16 require full sentences when describing transformations. Allow 60 minutes.

In the context of matrix arithmetic for a plotting task, calculate the sum of the 2x2 matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]].

(Total for Question 1 is 1 mark)

For a transformation exercise, calculate A - B where A = [[7, 2], [0, -3]] and B = [[4, -1], [5, 2]].

(Total for Question 2 is 1 mark)

A gardener uses a 2x1 column matrix to record a plant position. Calculate 4 times the column matrix v = [[2], [-1]] to scale coordinates for an enlargement.

(Total for Question 3 is 2 marks)

In matrix transformations, calculate the product of 2x2 matrices C = [[1, 2], [3, 0]] and D = [[-1, 4], [2, 1]] to combine two linear maps.

(Total for Question 4 is 2 marks)

Apply a linear transformation to a point. Calculate the image of point P with position vector p = [[3], [2]] under the 2x2 matrix T = [[2, 0], [1, -1]].

(Total for Question 5 is 2 marks)

For a 2x2 matrix used in area-scaling, calculate the determinant of M = [[5, 2], [1, 3]]. State the determinant value which indicates area scale factor.

(Total for Question 6 is 2 marks)

Find the inverse of the non-singular 2x2 matrix N = [[4, 1], [3, 2]] to reverse a linear transformation used on coordinates.

(Total for Question 7 is 3 marks)

Determine whether the 2x2 matrix S = [[2, 4], [1, 2]] is singular, for a coordinate transform used by Priya. If singular, give a brief reason.

(Total for Question 8 is 3 marks)

Calculate the product of a 2x2 transformation matrix and a column vector. Let U = [[0, -1], [1, 0]] (a rotation by 90 degrees) and q = [[4], [1]]. Find U q, the image of q under this rotation.

(Total for Question 9 is 2 marks)

Calculate the 2x2 matrix that represents a combined transformation of first scaling by 3 in all directions, then reflecting in the x-axis. Use matrices S = [[3, 0], [0, 3]] for enlargement and R = [[1, 0], [0, -1]] for reflection. Find the matrix product R S which maps a point after both transformations.

(Total for Question 10 is 2 marks)

Find the 2x2 matrix which represents a rotation of 180 degrees about the origin, and then use it to find the image of point with position vector r = [[-2], [5]].

(Total for Question 11 is 3 marks)

Combine two 2x2 linear transformations. Let A = [[0, 1], [1, 0]] (reflection in the line y = x) and B = [[2, 0], [0, 1]] (horizontal stretch by factor 2). Calculate the single 2x2 matrix that is equivalent to doing B then A (i.e. compute A B).

(Total for Question 12 is 4 marks)

Describe fully the single transformation that maps the square with vertices at (1,1), (2,1), (2,2), (1,2) to the square with vertices at ( -2, -2 ), ( -4, -2 ), ( -4, -4 ), ( -2, -4 ). Give the transformation as a combination of standard terms (scale factor, centre, rotation angle, reflection as appropriate) and any translation vector.

(Total for Question 13 is 4 marks)

Describe fully the single transformation that maps point A with position vector a = [[1], [0]] to A' with position vector [[0], [2]], and maps B with position vector b = [[0], [1]] to B' with position vector [[-2], [0]]. Use standard transformation language (rotation, reflection, enlargement, translation) and state centre or angle as needed.

(Total for Question 14 is 5 marks)

Two transformations are combined: first reflect in the y-axis, then translate by vector t = [[3], [1]]. Describe fully the single transformation that maps an arbitrary point (x,y) to its final image. Give the type of transformation, and if it is not a pure single linear transformation explain it as 'reflection then translation' and give the net effect on coordinates.

(Total for Question 15 is 5 marks)

A map on the plane is represented by the matrix P = [[-1, 0], [0, 1]] followed by the translation vector u = [[2], [0]]. Describe fully the single transformation performed, and give the image of point (3, -1).

(Total for Question 16 is 5 marks)

Mark scheme · IG.M5 Matrices and Matrix Transformations

Question 1

B1 A + B = [[3, 4], [1, 4]] cao
Answer: [[3, 4], [1, 4]]

Question 2

B1 A - B = [[3, 3], [-5, -5]] cao
Answer: [[3, 3], [-5, -5]]

Question 3

M1 multiplies both entries by 4
A1 4v = [[8], [-4]] cao
Answer: [[8], [-4]]

Question 4

M1 multiplies rows of C by columns of D to form product entries
A1 CD = [[3, 6], [-3, 12]] cao
Answer: [[3, 6], [-3, 12]]

Question 5

M1 performs matrix multiplication T p
A1 T p = [[6], [1]] cao
Answer: [[6], [1]]

Question 6

M1 uses ad - bc method correctly
A1 determinant = 13 cao
Answer: 13

Question 7

M1 calculates determinant correctly as 5
M1 forms adjugate matrix by swapping diagonal entries and negating off-diagonal entries
A1 N^{-1} = (1/5)[[2, -1], [-3, 4]] cao
Answer: (1/5)[[2, -1], [-3, 4]]

Question 8

M1 calculates determinant correctly using ad - bc
M1 states determinant = 0 or shows rows are proportional
A1 concludes S is singular with correct reason, e.g. det = 0 so no inverse exists
Answer: Singular, because determinant = 2*2 - 4*1 = 4 - 4 = 0

Question 9

M1 performs correct multiplication of U and q
A1 U q = [[-1], [4]] cao
Answer: [[-1], [4]]

Question 10

M1 multiplies R and S in correct order R S
A1 R S = [[3, 0], [0, -3]] cao
Answer: [[3, 0], [0, -3]]

Question 11

M1 states or writes rotation by 180 degrees matrix as [[-1, 0], [0, -1]]
M1 multiplies matrix by the column vector r
A1 image = [[2], [-5]] cao
Answer: [[2], [-5]]

Question 12

M1 multiplies A and B in the correct order A B
M1 computes entries correctly for first row
M1 computes entries correctly for second row
A1 A B = [[0, 1], [2, 0]] cao
Answer: [[0, 1], [2, 0]]

Question 13

M1 identifies change in size: scale factor 2 or 2x enlargement or reduction
M1 identifies orientation change: rotation by 180 degrees or reflection equivalent
M1 gives correct centre or translation vector, e.g. maps centre (1.5,1.5) to centre (-3, -3) so translation vector is (-4.5, -4.5) or states combined as enlargement with negative scale about origin
A1 fully correct description, for example: enlargement by scale factor 2 with centre at the origin combined with a reflection through the origin, equivalent to a rotation by 180 degrees and then scale factor 2, which maps the small square to the large one; or concise equivalent description with correct translation vector

Question 14

M1 identifies mapping of basis vectors implies linear map corresponding to matrix [[0, -2], [2, 0]] or recognises rotation and scale
M1 shows that a maps to (0,2) and b maps to (-2,0) consistent with a 90 degree rotation combined with scale factor 2
M1 states the centre is the origin and gives angle and scale factor, e.g. rotation by 90 degrees anticlockwise and enlargement by factor 2 about the origin
M1 explains that this is equivalent to multiplying by matrix [[0, -2], [2, 0]] or alternatively enlargement by 2 then rotation by 90 degrees
A1 full correct description, e.g. 'enlargement by factor 2 with centre at the origin combined with a rotation of 90 degrees anticlockwise about the origin' or equivalent wording

Question 15

M1 describes reflection in y-axis as mapping (x,y) to (-x,y)
M1 applies translation to give (-x + 3, y + 1)
M1 states that the combined map is not a single linear transformation but an isometry composed of reflection then translation
M1 gives final coordinate rule correctly as (x,y) -> (3 - x, y + 1)
A1 full clear description that the mapping is: reflect in the y-axis, then translate by vector (3,1), equivalently (x,y) maps to (3 - x, y + 1)

Question 16

M1 identifies P as reflection in the y-axis
M1 applies reflection to point (3, -1) to get (-3, -1)
M1 applies translation vector u to give final image (-1, -1)
M1 states combined transformation fully as reflection in the y-axis then translation by (2,0) and notes it is not a single 2x2 linear map
A1 final image given and description complete, e.g. image of (3, -1) is (-1, -1) and the transformation is reflect in the y-axis then translate by (2,0)

Matrices and Matrix Transformations - Worksheets, Questions and Revision