Theory question 110
Matrices and DeterminantsThree linear transformations, P, Q and R in the oxy plane are defined by
P: (x, y) → (−4x − y, 2x)
Q: (x, y) → (y, 6x − 9y)
R: (x, y) → (x − 2y, 3x + 5y)
(a) Write down the matrices of P, Q and R. (b) Find: (i) 2P − 3R + Q; (ii) QR; (iii) the inverse of the matrix R.
Show the model answer
Model answer
Matrices written row by row:
(a) P = (−4 −1 / 2 0), Q = (0 1 / 6 −9), R = (1 −2 / 3 5).
(b)(i) 2P − 3R + Q = (−8 −2 / 4 0) − (3 −6 / 9 15) + (0 1 / 6 −9) = (−11 5 / 1 −24).
(ii) QR = (0 1 / 6 −9)(1 −2 / 3 5) = (3 5 / −21 −57).
(iii) |R| = 5 + 6 = 11; R⁻¹ = (1/11)(5 2 / −3 1).
Explanation
Each row of the matrix holds the coefficients of x and y in one output coordinate.