Theory question 111
Matrices and DeterminantsThe images of points (2, 3) and (4, 5) under a linear transformation A are (3, 4) and (5, 6) respectively. Find the:
(a) matrix A;
(b) inverse of A;
(c) point whose image is (−1, 1).
Show the model answer
Model answer
(a) Let A = (a b / c d) (row by row): 2a + 3b = 3, 4a + 5b = 5 give b = 1, a = 0
2c + 3d = 4, 4c + 5d = 6 give d = 2, c = −1
A = (0 1 / −1 2)
(b) |A| = 0 × 2 − 1 × (−1) = 1; A−1 = (2 −1 / 1 0)
(c) A−1(−1 / 1) = (2(−1) − 1(1) / 1(−1) + 0) = (−3 / −1); the point is (−3, −1)
Explanation
Set up two pairs of simultaneous equations from the images; use the inverse to go backwards.