This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Set up the matrix multiplication for points A and B. Let the transformation matrix be M = a & b \\ c & d . The transformation maps an original point (x, y) to an image point (x', y') such that: a & b \\ c & d x \\ y = x' \\ y' For point A (1,2) mapping to A' ((1)/(2), 1): a & b \\ c & d 1 \\ 2 = (1)/(2) \\ 1 This gives the equations: 1) a(1) + b(2) = (1)/(2) a + 2b = (1)/(2) 2) c(1) + d(2) = 1 c + 2d = 1 For point B (4,2) mapping to B' (2,1): a & b \\ c & d 4 \\ 2 = 2 \\ 1 This gives the equations: 3) a(4) + b(2) = 2 4a + 2b = 2 4) c(4) + d(2) = 1 4c + 2d = 1 Step 2: Solve the system of equations for a and b. We use equations (1) and (3): a + 2b = (1)/(2) (1) 4a + 2b = 2 (3) Subtract equation (1) from equation (3): (4a + 2b) - (a + 2b) = 2 - (1)/(2) 3a = (4)/(2) - (1)/(2) 3a = (3)/(2) a = (3)/(2) × (1)/(3) a = (1)/(2) Substitute a = (1)/(2) into equation (1): (1)/(2) + 2b = (1)/(2) 2b = 0 b = 0 Step 3: Solve the system of equations for c and d. We use equations (2) and (4): c + 2d = 1 (2) 4c + 2d = 1 (4) Subtract equation (2) from equation (4): (4c + 2d) - (c + 2d) = 1 - 1 3c = 0 c = 0 Substitute c = 0 into equation (2): 0 + 2d = 1 2d = 1 d = (1)/(2) Step 4: Form the transformation matrix M. Using the values a = (1)/(2), b = 0, c = 0, d = (1)/(2): M = (1)/(2) & 0 \\ 0 & (1)/(2) The transformation matrix is (1)/(2) & 0 \\ 0 & (1)/(2) . What's next?