Theory question 112
Matrices and DeterminantsThe images of (3, 2) and (−1, 4) under a linear transformation T are (−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 transformations T and P.
(b) Calculate TP.
(c) Find the image of the point x(4, 3) under TP.
Show the model answer
Model answer
(a) Let T = (a b / c d): 3a + 2b = −1, −a + 4b = 7 give a = −9/7, b = 10/7
3c + 2d = 4, −c + 4d = 11 give c = −3/7, d = 37/14
T = (−9/7 10/7 / −3/7 37/14); P = (1 1 / 1 2) (matrices row by row)
(b) TP = (1/7 11/7 / 31/14 34/7)
(c) TP(4, 3): (4/7 + 33/7, 62/7 + 102/7) = (37/7, 164/7)
Explanation
Write the images as matrix equations and solve two pairs of simultaneous equations.