![Given matrices A = [[3, 1], [6, 2]] and B = [[-1, 3], [3, 9]], calculate (A+B)A. Also, given 2M + 3AB = 2CB - BA, express M in terms of CB and AB.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1774859951404-3f09f7122e4384bc.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: Rearrange the given equation to solve for M. The given equation is 2M + 3AB = 2CB - BA. Subtract 3AB from both sides: 2M = 2CB - BA - 3AB Step 2: Use the vector property BA = -AB. Substitute -AB for BA in the equation: 2M = 2CB - (-AB) - 3AB 2M = 2CB + AB - 3AB Step 3: Combine like terms. 2M = 2CB - 2AB Step 4: Divide by 2 to find M. M = CB - AB Step 5: Express CA using vector addition. Using the triangle rule for vector addition, CA can be expressed as the sum of vectors from C to B and B to A: CA = CB + BA Step 6: Substitute BA = -AB into the expression for CA. CA = CB + (-AB) CA = CB - AB Step 7: Compare the expressions for M and CA. From Step 4, we have M = CB - AB. From Step 6, we have CA = CB - AB. Since both expressions are identical, we have proven that M = CA. Therefore, M = CA.