Theory question 110
Matrices and Determinants(a) If A = (−2 5 / 4 3) and B = (3 1 / 2 3) find the value of x and y such that BA = 2(3 7 / −2 x) + (y 4 / 12 −3). (Matrices are written row by row.)
(b) Two functions, f and g are defined by f : x → ½x + 1 and g : x → (5x − 1)/3. Find (i) g−1 (ii) g−1 ∘ f.
Show the model answer
Model answer
(a) BA = (3(−2) + 1(4) 3(5) + 1(3) / 2(−2) + 3(4) 2(5) + 3(3)) = (−2 18 / 8 19)
Right side = (6 + y 18 / 8 2x − 3)
6 + y = −2, so y = −8; 2x − 3 = 19, so x = 11
(b)(i) y = (5x − 1)/3 gives x = (3y + 1)/5, so g−1(x) = (3x + 1)/5
(ii) g−1(f(x)) = [3(½x + 1) + 1]/5 = (3x + 8)/10
Explanation
Multiply B by A (rows of B into columns of A), then compare corresponding entries.