The images of (3,2) and (-1,4) under a linear transformation
T and (-1,4) and (7, 11) respectively. P is another transformation where
P: (x, y) - (x + y, x + 2y).
(a) Find the matrices T and P of the linear transformation T and P.
(b) Calculate TP.
(c) Find the image of the point X (4, 3) under TP.
The chief examiner stated that most candidates did not attempt this question.
Candidates were expected to show that ifT = [~ ~J then [~ ~~] = ~lJ
and (~ ~ J~ 1 J =[;lJ
i.e. 3a + 2b = -1 ............................... (i)
3c + 2d 4 ,,·······""· .. ········(ii)
-a + 4b 7 ......................... (iii)
-c + 4d 11 ........................ (v)
Solving (i) and (iii) simultaneously gave a = -9 and b = 10 . Similarly, solving (ii) 7 7
and (iv) simultaneously gave c = -3 and d = 37 . Thus the transformations are
-9 10J 7 [_~4 1~ [ J ~1 1~
T = !3 ;7 ' P = ~ ~J. Hence TP = !3 ;7i ~ = 11 ;4
7 14 ~1 1~ 7 14 14 7
The image ofx(4,3) under TP = ~ ;4 [~ JWhiCh gave (3;, 1~4).
14 7
