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
Hey Ria, good to see you again. Here's how to solve the problem: a) Find (A), the determinant of A Step 1: Calculate the determinant of matrix A. A = 2 & 1 & 2 \\ 3 & 1 & 2 \\ 2 & 2 & 1 The determinant is calculated as: (A) = 2 1 & 2 \\ 2 & 1 - 1 3 & 2 \\ 2 & 1 + 2 3 & 1 \\ 2 & 2 (A) = 2((1)(1) - (2)(2)) - 1((3)(1) - (2)(2)) + 2((3)(2) - (1)(2)) (A) = 2(1 - 4) - 1(3 - 4) + 2(6 - 2) (A) = 2(-3) - 1(-1) + 2(4) (A) = -6 + 1 + 8 (A) = 3 b) Find A^-1, the inverse of A Step 2: Find the cofactor matrix of A. The cofactor C_ij is (-1)^i+j times the determinant of the submatrix obtained by removing row i and column j. C_11 = 1 & 2 \\ 2 & 1 = 1 - 4 = -3 C_12 = - 3 & 2 \\ 2 & 1 = -(3 - 4) = 1 C_13 = 3 & 1 \\ 2 & 2 = 6 - 2 = 4 C_21 = - 1 & 2 \\ 2 & 1 = -(1 - 4) = 3 C_22 = 2 & 2 \\ 2 & 1 = 2 - 4 = -2 C_23 = - 2 & 1 \\ 2 & 2 = -(4 - 2) = -2 C_31 = 1 & 2 \\ 1 & 2 = 2 - 2 = 0 C_32 = - 2 & 2 \\ 3 & 2 = -(4 - 6) = 2 C_33 = 2 & 1 \\ 3 & 1 = 2 - 3 = -1 The cofactor matrix is: C = -3 & 1 & 4 \\ 3 & -2 & -2 \\ 0 & 2 & -1 Step 3: Find the adjugate matrix and the inverse A^-1. The adjugate matrix is the transpose of the cofactor matrix: adj(A) = C^T = -3 & 3 & 0 \\ 1 & -2 & 2 \\ 4 & -2 & -1 The inverse matrix A^-1 is (1)/((A)) adj(A): A^-1 = (1)/(3) -3 & 3 & 0 \\ 1 & -2 & 2 \\ 4 & -2 & -1 A^-1 = -1 & 1 & 0 \\ (1)/(3) & -(2)/(3) & (2)/(3) \\ (4)/(3) & -(2)/(3) & -(1)/(3) Solve the system of equations Step 4: Write the system of equations in matrix form AX = B. The given system is: 2x + y + 2z = 3 3x + y + 2z = 3 2x + 2y + z = 2 This can be written as: 2 & 1 & 2 \\ 3 & 1 & 2 \\ 2 & 2 & 1 x \\ y \\ z = 3 \\ 3 \\ 2 Here, A = 2 & 1 & 2 \\ 3 & 1 & 2 \\ 2 & 2 & 1 , X = x \\ y \\ z , and B = 3 \\ 3 \\ 2 . Step 5: Solve for X using X = A^-1B. x \\ y \\ z = (1)/(3) -3 & 3 & 0 \\ 1 & -2 & 2 \\ 4 & -2 & -1 3 \\ 3 \\ 2 Perform the matrix multiplication: x \\ y \\ z = (1)/(3) (-3)(3) + (3)(3) + (0)(2) \\ (1)(3) + (-2)(3) + (2)(2) \\ (4)(3) + (-2)(3) + (-1)(2) x \\ y \\ z = (1)/(3) -9 + 9 + 0 \\ 3 - 6 + 4 \\ 12 - 6 - 2 x \\ y \\ z = (1)/(3) 0 \\ 1 \\ 4 x \\ y \\ z = 0 \\ (1)/(3) \\ (4)/(3) The solution to the system of equations is: x = 0, y = (1)/(3), z = (4)/(3) What's next? Send 'em! 📸