Theory question 110
Matrices and Determinants(a) Write down the matrix A of the linear transformation A(x, y) → (2x − y, −5x + 3y).
(b) If B = (3 1 / 5 2), find (i) A2 − B2; (ii) matrix C = B2A (iii) the point M(x, y) whose image under the linear transformation C is M(10, 18).
(c) What is the relationship between matrix A and matrix C? (Matrices are written row by row.)
Show the model answer
Model answer
(a) A = (2 −1 / −5 3)
(b)(i) A2 = (9 −5 / −25 14), B2 = (14 5 / 25 9)
A2 − B2 = (−5 −10 / −50 5)
(ii) C = B2A = (14 5 / 25 9)(2 −1 / −5 3) = (3 1 / 5 2)
(iii) 3x + y = 10 and 5x + 2y = 18 give x = 2, y = 4; M(2, 4)
(c) AC = (2 −1 / −5 3)(3 1 / 5 2) = (1 0 / 0 1) = I, so C is the inverse of A (C = A−1, which here equals B)
Explanation
Multiply row by column carefully; check (c) by working out AC.