Theory question 110
Matrices and DeterminantsP and Q are two linear transformations in the x − y plane defined by P: (x, y) → (−3x + 6y, 4x + y) and Q: (x, y) → (2x − 3y, −4x − 6y). (a) Write down the matrices of P and Q. (b) What is the image of (−2, −3) under the transformation Q? (c) Obtain a single transformation representing the transformation Q followed by P. (d) Find the image of (1, 4) when transformed by Q followed by P. (e) Find the image P1 of the point (−√2, 2√2) under an anticlockwise rotation of 225° about the origin.
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Matrices written row by row:
(a) P = (−3 6 / 4 1), Q = (2 −3 / −4 −6).
(b) Q(−2, −3) = (−4 + 9, 8 + 18) = (5, 26).
(c) Q followed by P is PQ = (−30 −27 / 4 −18).
(d) PQ(1, 4) = (−30 − 108, 4 − 72) = (−138, −68).
(e) Rotation matrix (cos 225° −sin 225° / sin 225° cos 225°) = (−√2/2 √2/2 / −√2/2 −√2/2).
Image = (1 + 2, 1 − 2) = (3, −1).
'Q followed by P' means apply Q first, so the matrix is PQ.