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
Given the matrices: A = 5 & 9 \\ -2 & 4 B = 2 & 3 \\ 4 & 1 (i) Evaluate AB Step 1: Multiply matrix A by matrix B. AB = 5 & 9 \\ -2 & 4 2 & 3 \\ 4 & 1 Step 2: Perform the matrix multiplication. AB = (5)(2) + (9)(4) & (5)(3) + (9)(1) \\ (-2)(2) + (4)(4) & (-2)(3) + (4)(1) AB = 10 + 36 & 15 + 9 \\ -4 + 16 & -6 + 4 AB = 46 & 24 \\ 12 & -2 The result for AB is: 46 & 24 \\ 12 & -2 (ii) Evaluate BA Step 1: Multiply matrix B by matrix A. BA = 2 & 3 \\ 4 & 1 5 & 9 \\ -2 & 4 Step 2: Perform the matrix multiplication. BA = (2)(5) + (3)(-2) & (2)(9) + (3)(4) \\ (4)(5) + (1)(-2) & (4)(9) + (1)(4) BA = 10 - 6 & 18 + 12 \\ 20 - 2 & 36 + 4 BA = 4 & 30 \\ 18 & 40 The result for BA is: 4 & 30 \\ 18 & 40 (iii) Evaluate A(AB) Step 1: Use the result of AB from part (i). AB = 46 & 24 \\ 12 & -2 Step 2: Multiply matrix A by the result of AB. A(AB) = 5 & 9 \\ -2 & 4 46 & 24 \\ 12 & -2 Step 3: Perform the matrix multiplication. A(AB) = (5)(46) + (9)(12) & (5)(24) + (9)(-2) \\ (-2)(46) + (4)(12) & (-2)(24) + (4)(-2) A(AB) = 230 + 108 & 120 - 18 \\ -92 + 48 & -48 - 8 A(AB) = 338 & 102 \\ -44 & -56 The result for A(AB) is: 338 & 102 \\ -44 & -56