MATRICES EXAM QUESTIONS

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1 MATRICES EXAM QUESTIONS (Part One)

2 Question 1 (**) The matrices A, B and C are given below in terms of the scalar constants a, b, c and d, by 2 3 A =, 1 a b 1 B =, c C =. d 4 Given that A + B = C, find the value of a, b, c and d. a = 8, b = 3, c = 2, d = 3 Question 2 (**) Find, in terms of k, the inverse of the following 2 2 matrix. Verify your answer by multiplication. k k + 1 M =. k + 1 k + 2 FP1-M, 1 k 2 k + 1 M = k + 1 k

3 Question 3 (**) The 2 2 matrices A, B and C are given below in terms of the scalar constants a, b and c. a 2 A =, B =, b 2 1 c C =. 3 2 Given that 2A 3B = 4C, find the value of a, b and c. a = 1, b = 2, c = 2

4 Question 4 (**) The 2 2 matrix A represents a rotation by 90 anticlockwise about the origin O. The 2 2 matrix B represents a reflection in the straight line with equation y = x. a) Write down the matrices A and B. The 2 2 matrix C represents a rotation by 90 anticlockwise about the origin O, followed by a reflection about the straight line with equation y = x. b) Find the elements of C. c) Describe geometrically the transformation represented by C. 0 1 A =, B =, C =, reflection in the y axis 0 1

5 Question 5 (**) The 2 2 matrix A, is defined as where a and b are constants. 2 a A = b 2 The matrix A, maps the point P ( 2,5) onto the point Q( 1,2). a) Find the value of a and the value of b. A triangle T 1 with an area of 9 square units is transformed by A into the triangle T 2. b) Find the area of T 2. a = 1, b = 6, area = 18

6 Question 6 (**) The 2 2 matrices A, B and C are given below in terms of the scalar constants x. 2 x A =, B = and 1 4 3x C =. 7 x 7 a) Find an expression for AB, in terms of x. b) Determine the value of x, given T B A T = C. 4 + x 2 + 4x AB =, x = Question 7 (**) The 2 2 matrices A and B are given by 5 2 A = and B =. 4 5 Find the 2 2 matrix X that satisfy the equation AX = B. 1 2 X = 2 1

7 Question 8 (**) The triangle T 1 is mapped by the 2 2 matrix onto another triangle 2 C2 ( 18, 14). Find the coordinates of the vertices of T A = 3 1 T, whose vertices have coordinates A ( ), ( ) 2 1,2 B 2 10,15 and A 1( 1,1 ), B 1 ( 4, 3), C1 ( 2,8 )

8 Question 9 (**) A plane transformation maps the general point ( x, y ) onto the general point (, ) X Y, by where A is the 2 2 matrix X x = Y A y, a) Give a geometrical description for the transformation represented by A, stating the equation of the line of invariant points under this transformation b) Calculate 2 A and describe geometrically the transformation it represents. shear parallel to y = 0, ( 0,1) ( 2,1) line of invariant points y = 0, ( ) ( ) shear parallel to y = 0, 0,1 4,1

9 Question 10 (**) The triangle T 1 is mapped by the 2 2 matrix 4 1 B = 3 1 onto a triangle T 2, whose vertices are the points with coordinates A 2 ( 4,3 ), B 2 ( 4,10) and C 2 ( 16,12 ). a) Find the coordinates of the vertices of T 1. b) Determine the area of T 2. A 1( 1,0 ), B 1 ( 2,4), 1 ( 4,0) C, area = 42

10 Question 11 (**) The 2 2 matrix C is defined, in terms of a scalar constant a, by 3 a C =. 5 2 a) Determine the value of a, given that C is singular. The 2 2 matrix D is given by b) Find the inverse of D. 2 1 D =. 4 3 The point P is transformed by D onto the point Q( 6k 1,14k 1) constant. c) Determine, in terms of k, the coordinates of Q. + +, where k is a scalar a = 6, = D 2 2, Q( 2k + 1,2k 1)

11 Question 12 (**) A plane transformation maps the general point ( x, y ) to the general point (, ) X Y by X x =. Y y a) Find the area scale factor of the transformation. The points on a straight line which passes through the origin remain invariant under this transformation. b) Determine the equation of this straight line. SF = 16, y = 3 x 4 Question 13 (**) The distinct square matrices A and B are non singular. Simplify the expression, showing all steps in the workings. 1 ( ) 1 AB A B. 2 A

12 Question 14 (**) The distinct square matrices A and B have the properties 5 AB = B A and 6 = B I where I is the identity matrix. Prove that BAB = A. proof Question 15 (**) The 2 2 matrix A is given by 1 3 A =. 1 4 The 2 2 matrix B satisfies BA 2 = A. Find the elements of B. 4 3 B = 1 1

13 Question 16 (**+) The transformation represented by the 2 2 matrix A maps the point ( ) point ( 10,4 ), and the point ( 5, 2) onto the point ( 8, 2). 3,4 onto the Determine the elements of A. 2 1 A = 0 1 Question 17 (**+) The 2 2 matrix A is given below. Determine the elements of represented by A. 1 3 A = A and hence describe geometrically the transformation A =, 0 8 rotation of 120, anticlockwise & enlargement of S.F. 2, both about the origin and in any order.

14 Question 18 (**+) It is given that ( 2 1) A =, 3 B = and C =. 4 1 a) Determine the matrix AB. b) Find the elements of 2 BA 2C. ( 5) AB =, BA 2C = 2 13

15 Question 19 (**+) The 2 2 matrices A and B are given below 2 1 A = and B =. 2 2 The matrix C represents the combined effect of the transformation represented by the B, followed by the transformation represented by A. a) Determine the elements of C. b) Describe geometrically the transformation represented by C. 0 2 C =, enlargement by scale factor 2, reflection in the line y = x, in any order 2 0

16 Question 20 (**+) The 2 2 matrix D is given by 4 5 D =. 6 9 a) Given that I is the 2 2 identity matrix, show clearly that i. 2 D + 5D = 6I. ii. 1 = 1 ( + 5 ) D D I. 6 The transformation in the x-y plane, which is represented by the matrix D, maps the point P onto the point Q. The coordinates of Q are ( 7 2 k,9 6k ), where k is a constant. b) Determine, in terms of k, the coordinates of P. ( 2 + 3,2 + 1) Q k k

17 Question 21 (**+) The 2 2 matrix B maps the points with coordinates ( 1,2 ) and ( ) with coordinates ( 0,1 ) and ( 6, 1), respectively. 1,4 onto the points a) Find the elements of B. b) Determine whether B has an invariant line, or a line of invariant points. c) Describe geometrically the transformation represented by B. 2 1 B =, line of invariant points, y = x, shear 1 0

18 Question 22 (***) The 2 2 matrices A and B are given by 5 7 A = ; B = Find the 2 2 matrix X that satisfy the equation AX = B 1 3 X = 2 3 Question 23 (***) It is given that A and B are 2 2 matrices that satisfy ( ) 1 AB and ( ) det = 18 det B = 3. A square S, of area 6 2 cm, is transformed by A to produce an image S. Given that S is also a square, determine its perimeter. 72 cm

19 Question 24 (***) The 2 2 matrix A is given by where a and b are scalar constants. 2 a A =, 3 b a) If the point with coordinates ( 1,1 ) is mapped by A onto the point with coordinates ( 1,3 ), determine the value of a and the value of b. b) Show that 2 A = 2A 3I. The inverse of A is denoted by c) Use part (b) to show further that 1 A and I is the 2 2 identity matrix. i. 3 A = A 6I. 1 ii. A = 1 ( 2 I A ) 3 FP1-N, a = 1, b = 0

20 Question 25 (***) A transformation in the x-y plane is represented by the 2 2 matrix 1 3 T =. 1 4 A quadrilateral Q has vertices at the points with coordinates ( 20,6 ), ( 26,9 ), ( 44,12 ) and ( 50,15 ). These coordinates are given in cyclic order. The vertices of Q are transformed by T. a) Find the positions of the vertices of the image of Q. b) Determine the area of Q, fully justifying your reasoning. ( 2,4 ), ( 4,10 ), ( 8,4 ), ( 5,10 ), area = 21

21 Question 26 (***) The 2 2 matrix B is given by where a and b are scalar constants. a 2 B =, 3 b The point with coordinates ( 3,1 ) is mapped by B onto the point with coordinates ( 5,13 ). a) Determine the value of a and the value of b. The inverse of B is denoted by b) Show that 1 B and I is the 2 2 identity matrix. 2 B = 5B + 2I. c) Show further that i. 3 B = 27B + 10I. 1 ii. B = 1 ( B 5I ) 2 a = 1, b = 4

22 Question 27 (***) A transformation in the x-y plane consists of......a reflection about the line with equation y = x... followed by an anticlockwise rotation about the origin by followed by a reflection about the x axis. Use matrices to describe geometrically the resulting combined transformation. FP1-O, rotation about the origin by 180

23 Question 28 (***+) The 2 2 matrices A and B are defined by 4 1 A = and B =. 3 5 a) Find 1 A, the inverse of A. b) Find a matrix C, so that ( ) 1 B + C = A. c) Describe geometrically the transformation represented by C. SYNF-B, A =, C =, rotation by

24 Question 29 (***+) The 2 2 matrices 0 1 A = and B =, 0 1 represent linear transformations in the x-y plane. a) Give full geometrical descriptions for each of the transformations represented by A and B. The figure below shows a right angled triangle T, with vertices at the points A( 1, 1), B( 3, 1) and C ( 3,3). y C ( 3,3) O T x A( 1, 1) B( 3, 1) The triangle T is first transformed by A and then by B, producing the triangle T. b) Find the single matrix that represents this composite transformation. c) Determine the new coordinates of the vertices of T. d) Calculate the area of T. [continues overleaf]

25 [continued from overleaf] The triangle T is then reflected in the straight line with equation y = x to give the triangle T. e) Find the single matrix that maps T back onto T. SYNF-D, rotation, 90, clockwise, enlargement, in x only,scale factor 2, 0 2 = 1 0 BA, A ( 2, 1 ), B ( 2, 3 ), C ( 6, 3), area = 8,

26 Question 30 (***+) The 2 2 matrix A given below, represents a transformation in the x-y plane. 0 1 A =. 1 0 a) Describe geometrically the transformation represented by A. The transformation described by A is equivalent to a reflection about the straight line with equation y = x, followed by another transformation described by the matrix C. b) Find the matrix C, and describe it geometrically. rotation by 90, anticlockwise, 1 0 C =, reflection about the x axis 0 1

27 Question 31 (***+) The 2 2 matrix M is defined by 0 3 M =. 3 0 Find, by calculation, the equations of the two lines which pass through the origin, that remain invariant under the transformation represented by M. SYNF-A, y = ± x

28 Question 32 (***+) Find the image of the straight line with equation 2x + 3y = 10, under the transformation represented by the 2 2 matrix 1 2 A =. 3 1 SYNF-E, 11x + y = 70

29 Question 33 (****) The 2 2 matrix R represents a reflection where the point ( 2,1 ) gets mapped onto the point ( 6, 5), and the line with equation y = 1 x is a line of invariant points. 2 a) Determine the elements of R. The 2 2 matrix M represents the combined transformation of the reflection represented by R, followed by another transformation T M = b) Given that T is also a reflection determine, in exact simplified form, the equation of the line of reflection of T. FP1-X, 2 2 R = 3 2, 1 2 x = 1 7 y = 1 2 z 2

30 Question 34 (****) Under the transformation represented by the 2 2 matrix the straight line with equation y 1 2 A =, 4 7 = mx is reflected about the x axis. Find the possible values of m. FP1-V, m = 1, m = 2

31 Question 35 (****) The 2 2 matrix A is given below. 3 2 A =. 7 5 a) Find scalar constants, k and h, so that 2 A + ki = ha. b) Use part (a) to determine No credit will be given for finding 1 A, the inverse of A. 1 A by a direct method. FP1-W, k = 1, h = 8, A = 7 3

32 Question 36 (****) The 2 2 matrix A given below represents a transformation in the x-y plane. 3 1 A =. 5 2 The straight line L with equation y = 2x + 1 is transformed by A into the straight line L. a) Find a Cartesian equation of L. The straight line M is transformed by A into the straight line M with equation 11x + 6y = 4. b) Find a Cartesian equation of M. L : y = 1 x, M : y = 4 3x

33 Question 37 (****) Describe fully the transformation given by the following 2 2 matrix The description must be supported by mathematical calculations. reflection in y = 2x

34 Question 38 (****) A composite transformation in the x-y plane consists of i. a uniform enlargement about the origin of scale factor k, k > 0, denoted iii by the matrix E. ii. a shear parallel to the straight line L, denoted by the matrix S. It is given that ES = SE = 9 36 a) Show clearly that k = 24. b) Find a Cartesian equation of L. y = 3 x 4

35 Question 39 (****) A plane transformation maps the general point ( x, y ) onto the general point (, ) X Y, by X 2 1 x =. Y 1 2 y a) Find the area scale factor of the transformation. b) Determine the equation of the straight line of invariant points under this transformation. c) Show that all the straight lines with equation of the form x + y = c, where c is a constant, are invariant lines under this transformation. d) Hence describe the transformation geometrically. SF = 3, y = x, stretch perpendicular to the line y = x, by area scale factor 3

36 Question 40 (****) A transformation 2 2 T : R R is represented by the following 2 2 matrix. 3 8 A =. 1 3 a) Find the determinant of A and explain its significance with reference to its sign and its magnitude. b) Find the equation of the straight line of the invariant points under the transformation represented by A. c) Determine the entries of the 2 2 matrix B which represents a reflection about the straight line found in part (b), giving all its entries as simple fractions. The transformation represented by A, consists of a shear represented by the matrix C, followed by a reflection represented by the matrix B. d) Determine the matrix C and describe the shear. det A = 1, y = 1 x, B =, C =

37 Question 41 (****+) A transformation T, maps the general point ( x, y ) onto the general point (, ) X Y, by X 1 2 x =. Y 2 3 y a) Find the area scale factor of the transformation. b) Determine the equation of the line of invariant points under this transformation. c) Show that all the straight lines of the form y = x + c, where c is a constant, are invariant lines under T. d) Hence state the name of T. e) Show that the acute angle formed by the straight line with equation y = x and its the image under T is 3π 5 arctan 4 3. SF = 1, y = x, shear

38 Question 42 (****+) A curve has equation 2 2 5x 16xy + 13y = 25. This curve is the image of another curve C, under the transformation defined by the 2 2 matrix A, given below. 1 2 A =. 2 3 Show that the equation ofc is the circle with equation x y = 25. FP1-U, proof

39 Question 43 (****+) The 2 2 matrix P is given below. 2 1 P = 3 2 The points on the x-y plane which lie on a curve C are transformed by P onto the points which lie on the curve with equation x 16xy + 5y + 8x 6y = 4. Determine an equation for C and hence describe it geometrically. 2 2 SPX-B, ( x ) ( y ) = 9

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mathematics SKE, Strand J STRAND J: TRANSFORMATIONS, VECTORS and MATRICES J4 Matrices Text Contents * * * * Section J4. Matrices: Addition and Subtraction J4.2 Matrices: Multiplication J4.3 Inverse Matrices: