EdExcel FP3 Matrices - GlynMathsAlevel

EdExcel Further Pure 1 Matrix algebra Chapter Assessment  3 2  1. The matrix   represents a transformation, T. The inverse of transformation  2 1  T is W. (i) Find the matrix representing W. [3] (ii) Show that TW = WT = I. [3] 2. A transformation T is given by:  x   x  2y  T:       y  y  x  (i) Write down the matrix representing the transformation T. [2] (ii) The transformation S is an anticlockwise rotation through 90° about the origin. Find the matrix representing the transformation S. [2] (iii) Find the single matrix representing the transformation S followed by the transformation T. [3]

 6 3  3. The plane is transformed by means of the matrix M =  .  4 k  (i) In the case where k = 3: (a) find the determinant of M, (b) find the area of the image of a triangle with area 4 square units. (ii) Find the value of k for which the matrix is singular.

[2] [2] [2]

 2 1 1 0  4. The matrix A    and the matrix B   . 3 1   3 2  (i) Find A-1 and B-1. (ii) Hence or otherwise, find (AB)-1.

[4] [3]

 2  3  . 5. Matrix A represents a transformation T where A =   4  4 (i) Find the image of the point (2, -1) under the transformation T. (ii) Find the inverse of A. (iii) Find the coordinates of the point that is mapped to (9, 16) under transformation T. (iv) Find A2. d 0  stating the value of d. (v) Show that A3 =  0 d (vi) Give a geometrical description of the matrix A3.

[2] [3] [3] [2] [2] [2]

Total 40 marks

© MEI, 15/03/10

1/3

EdExcel FP1 Matrices Assessment solutions Matrix algebra Solutions to chapter assessment 1. (i)

 3 2  det    3  ( 4)  1  2 1   1 2  W  T 1     2 3  [3]

 3 2  1 2   1 0  (ii) TW      I  2 1  2 3   0 1   1 2  3 2   1 0  WT      I  2 3  2 1   0 1  [3] 2. (i) The point (1, 0) is mapped to the point (1, -1). The point (0, 1) is mapped to the point (2, 1).  1 2 The matrix representing T is therefore  .  1 1  [2] (ii) The point (1, 0) is mapped to the point (0, 1). The point (0, 1) is mapped to the point (-1, 0).  0 1  The matrix representing S is therefore  . 1 0  [2] (iii) S followed by T is represented by the matrix product TS.  1 2  0 1  TS      1 1  1 0   2 1    1 1  [3] 3. (i) (a) (b)

det M 

6 3  18  12  6 4 3

Area factor = 6, so area of image  4  6  24 square units.

[2] [2]

(ii) If the matrix is singular, then the determinant is zero. 6 3 det M   6k  12 4 k

det M  0  6k  12  0  k  2

© MEI, 09/06/08

[2] 2/3

EdExcel FP1 Matrices Assessment solutions 4. (i)

det A  (2  1)  ( 1  3)  2  3  5 A 1 

 1 1  3 2   

1 5

det B  (1  2)  (0  3)  2 B 1  

1  2 0  1  2 0      2  3 1  2  3 1 

(ii) (AB)-1  B 1 A 1

1 2 0  1  1 1     2  3 1  5  3 2  1 2 2    10  6 1  

5. (i)

 2 3  2   7   4 4  1    12       The image is (7, 12).

 2 3  (ii) det    8  12  4  4 4  1  4 3  A-1 =   4  4 2 

x  9 (iii) A       y   16  9  1  4 3  9  1  12   3  x 1   y   A  16    4 2  16    4    1      4   4     The point (3, -1) is mapped to (9, 16).  2 3  2 3   8 6  (iii) A² =    .  4 4  4 4   8 4   8 6  2 3   8 0  (iv) A³ =     so d = 8  8 4  4 4   0 8  (v) The matrix A³ is an enlargement, centre the origin, scale factor 8.

© MEI, 09/06/08

3/3