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: Let the given matrix be C. We are given that A is a symmetric matrix (A^T = A) and B is a skew-symmetric matrix (B^T = -B). The sum of these matrices is A+B = C. The given matrix C is: C = 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 The transpose of C is C^T: C^T = 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 Step 2: We know that A = (1)/(2)(C + C^T). First, calculate C + C^T: C + C^T = 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 + 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 = 2+2 & -2-1 & -4+1 \\ -1-2 & 3+3 & 4-2 \\ 1-4 & -2+4 & -3-3 = 4 & -3 & -3 \\ -3 & 6 & 2 \\ -3 & 2 & -6 Now, calculate A: A = (1)/(2) 4 & -3 & -3 \\ -3 & 6 & 2 \\ -3 & 2 & -6 = 2 & -3/2 & -3/2 \\ -3/2 & 3 & 1 \\ -3/2 & 1 & -3 Step 3: We know that B = (1)/(2)(C - C^T). First, calculate C - C^T: C - C^T = 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 - 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 = 2-2 & -2-(-1) & -4-1 \\ -1-(-2) & 3-3 & 4-(-2) \\ 1-(-4) & -2-4 & -3-(-3) = 0 & -1 & -5 \\ 1 & 0 & 6 \\ 5 & -6 & 0 Now, calculate B: B = (1)/(2) 0 & -1 & -5 \\ 1 & 0 & 6 \\ 5 & -6 & 0 = 0 & -1/2 & -5/2 \\ 1/2 & 0 & 3 \\ 5/2 & -3 & 0 Step 4: Evaluate the product AB. AB = 2 & -3/2 & -3/2 \\ -3/2 & 3 & 1 \\ -3/2 & 1 & -3 0 & -1/2 & -5/2 \\ 1/2 & 0 & 3 \\ 5/2 & -