![Given matrices A = [[3, 1], [6, 2]] and B = [[-1, 3], [3, 9]], calculate (A+B)A.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1774859436019-d2140451240e9540.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
Step 1: Calculate A+B. A+B = 3 & 1 \\ 6 & 2 + -1 & 3 \\ 3 & 9 = 3 + (-1) & 1 + 3 \\ 6 + 3 & 2 + 9 = 2 & 4 \\ 9 & 11 Step 2: Calculate (A+B)A. Let C = A+B. We need to calculate CA. CA = 2 & 4 \\ 9 & 11 3 & 1 \\ 6 & 2 CA = (2)(3) + (4)(6) & (2)(1) + (4)(2) \\ (9)(3) + (11)(6) & (9)(1) + (11)(2) CA = 6 + 24 & 2 + 8 \\ 27 + 66 & 9 + 22 CA = 30 & 10 \\ 93 & 31 The final answer is 30 & 10 \\ 93 & 31 . Step 1: Express the left side of the equation using vector properties of a parallelogram. In a parallelogram ABCD, opposite sides are equal and parallel, so DC = AB. Substitute DC with AB in the left side of the equation: AB + AD + DC = AB + AD + AB AB + AD + DC = 2AB + AD (*) Step 2: Express the right side of the equation using vector properties and the midpoint definition. H is the midpoint of BC. This means BH = (1)/(2)BC. In a parallelogram ABCD, BC = AD. So, BH = (1)/(2)AD. Now, express AH using the triangle rule for vector addition: AH = AB + BH Substitute BH = (1)/(2)AD: AH = AB + (1)/(2)AD Step 3: Calculate 2AH. 2AH = 2(AB + (1)/(2)AD) 2AH = 2AB + 2((1)/(2)AD) 2AH = 2AB + AD (**) Step 4: Compare the results from Step 1 and Step 3. From (*) we have AB + AD + DC = 2AB + AD. From (**) we have 2AH = 2AB + AD. Since both sides simplify to the same expression, the equality is proven. Therefore, AB + AD + DC = 2AH.