![Given that matrices A = [[2, 1, 8], [4, 2, 0], [6, 7, 8]] and B = [[1, 1, 1], [1, 2, 1], [0, 4, 3]], Evaluate (i) A^T B^T (ii) (AB)^T](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1781007179785-b3662be5d0dfcd7b.png&w=3840&q=75)
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 = 2 & 1 & 8 \\ 4 & 2 & 0 \\ 6 & 7 & 8 and B = 1 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 4 & 3 I) Evaluate A^T B^T Step 1: Find the transposes of A and B. To find the transpose of a matrix, interchange its rows and columns. A^T = 2 & 4 & 6 \\ 1 & 2 & 7 \\ 8 & 0 & 8 B^T = 1 & 1 & 0 \\ 1 & 2 & 4 \\ 1 & 1 & 3 Step 2: Multiply A^T by B^T. A^T B^T = 2 & 4 & 6 \\ 1 & 2 & 7 \\ 8 & 0 & 8 1 & 1 & 0 \\ 1 & 2 & 4 \\ 1 & 1 & 3 A^T B^T = (2)(1)+(4)(1)+(6)(1) & (2)(1)+(4)(2)+(6)(1) & (2)(0)+(4)(4)+(6)(3) \\ (1)(1)+(2)(1)+(7)(1) & (1)(1)+(2)(2)+(7)(1) & (1)(0)+(2)(4)+(7)(3) \\ (8)(1)+(0)(1)+(8)(1) & (8)(1)+(0)(2)+(8)(1) & (8)(0)+(0)(4)+(8)(3) A^T B^T = 2+4+6 & 2+8+6 & 0+16+18 \\ 1+2+7 & 1+4+7 & 0+8+21 \\ 8+0+8 & 8+0+8 & 0+0+24 A^T B^T = 12 & 16 & 34 \\ 10 & 12 & 29 \\ 16 & 16 & 24 II) Evaluate (AB)^T Step 1: Calculate the product AB. AB = 2 & 1 & 8 \\ 4 & 2 & 0 \\ 6 & 7 & 8 1 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 4 & 3 AB = (2)(1)+(1)(1)+(8)(0) & (2)(1)+(1)(2)+(8)(4) & (2)(1)+(1)(1)+(8)(3) \\ (4)(1)+(2)(1)+(0)(0) & (4)(1)+(2)(2)+(0)(4) & (4)(1)+(2)(1)+(0)(3) \\ (6)(1)+(7)(1)+(8)(0) & (6)(1)+(7)(2)+(8)(4) & (6)(1)+(7)(1)+(8)(3) AB = 2+1+0 & 2+2+32 & 2+1+24 \\ 4+2+0 & 4+4+0 & 4+2+0 \\ 6+7+0 & 6+14+32 & 6+7+24 AB = 3 & 36 & 27 \\ 6 & 8 & 6 \\ 13 & 52 & 37 Step 2: Find the transpose of AB. (AB)^T = 3 & 6 & 13 \\ 36 & 8 & 52 \\ 27 & 6 & 37 (AB)^T = 3 & 6 & 13 \\ 36 & 8 & 52 \\ 27 & 6 & 37 That's 2 down. 3 left today — send the next one.