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
You're on a roll — the student's work for finding a and b is correct. I will complete the solution by finding c and d. Step 1: Set up the matrix multiplication. Given P = a & b \\ c & d , M = 3 & -2 \\ -4 & 3 , and N = 5 & 4 \\ -2 & 6 . The equation is PM = N. a & b \\ c & d 3 & -2 \\ -4 & 3 = 5 & 4 \\ -2 & 6 Step 2: Perform the matrix multiplication. (a)(3) + (b)(-4) & (a)(-2) + (b)(3) \\ (c)(3) + (d)(-4) & (c)(-2) + (d)(3) = 5 & 4 \\ -2 & 6 3a - 4b & -2a + 3b \\ 3c - 4d & -2c + 3d = 5 & 4 \\ -2 & 6 Step 3: Formulate systems of linear equations. Equating the corresponding elements gives two systems of equations: For a and b: 1) 3a - 4b = 5 2) -2a + 3b = 4 For c and d: 3) 3c - 4d = -2 4) -2c + 3d = 6 Step 4: Solve for a and b. Multiply equation (1) by 2 and equation (2) by 3: 2(3a - 4b) = 2(5) 6a - 8b = 10 3(-2a + 3b) = 3(4) -6a + 9b = 12 Add the two new equations: (6a - 8b) + (-6a + 9b) = 10 + 12 b = 22 Substitute b = 22 into equation (1): 3a - 4(22) = 5 3a - 88 = 5 3a = 5 + 88 3a = 93 a = (93)/(3) a = 31 Step 5: Solve for c and d. Multiply equation (3) by 2 and equation (4) by 3: 2(3c - 4d) = 2(-2) 6c - 8d = -4 3(-2c + 3d) = 3(6) -6c + 9d = 18 Add the two new equations: (6c - 8d) + (-6c + 9d) = -4 + 18 d = 14 Substitute d = 14 into equation (3): 3c - 4(14) = -2 3c - 56 = -2 3c = -2 + 56 3c = 54 c = (54)/(3) c = 18 Step 6: Write the matrix P. P = a & b \\ c & d = 31 & 22 \\ 18 & 14 The matrix P is: 31 & 22 \\ 18 & 14 Got more? Send 'em!