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.
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6. (a) Step 1: Write the given matrix A. A = 2 & 1 & 2 \\ 3 & 1 & 2 \\ 2 & 2 & 1 Step 2: Calculate the determinant of A, denoted as (A). For a 3 × 3 matrix a & b & c \\ d & e & f \\ g & h & i , the determinant is a(ei - fh) - b(di - fg) + c(dh - eg). (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 The determinant of A is 3. 6. (b) Step 1: Calculate the cofactor matrix of A. The cofactor C_ij is given by (-1)^i+j M_ij, where M_ij is the minor of the element a_ij. C_11 = + 1 & 2 \\ 2 & 1 = (1)(1) - (2)(2) = 1 - 4 = -3 C_12 = - 3 & 2 \\ 2 & 1 = -((3)(1) - (2)(2)) = -(3 - 4) = -(-1) = 1 C_13 = + 3 & 1 \\ 2 & 2 = (3)(2) - (1)(2) = 6 - 2 = 4 C_21 = - 1 & 2 \\ 2 & 1 = -((1)(1) - (2)(2)) = -(1 - 4) = -(-3) = 3 C_22 = + 2 & 2 \\ 2 & 1 = (2)(1) - (2)(2) = 2 - 4 = -2 C_23 = - 2 & 1 \\ 2 & 2 = -((2)(2) - (1)(2)) = -(4 - 2) = -(2) = -2 C_31 = + 1 & 2 \\ 1 & 2 = (1)(2) - (2)(1) = 2 - 2 = 0 C_32 = - 2 & 2 \\ 3 & 2 = -((2)(2) - (2)(3)) = -(4 - 6) = -(-2) = 2 C_33 = + 2 & 1 \\ 3 & 1 = (2)(1) - (1)(3) = 2 - 3 = -1 The cofactor matrix C is: C = -3 & 1 & 4 \\ 3 & -2 & -2 \\ 0 & 2 & -1 Step 2: Calculate the adjoint matrix of A, denoted as adj(A). The adjoint matrix is the transpose of the cofactor matrix. adj(A) = C^T = -3 & 3 & 0 \\ 1 & -2 & 2 \\ 4 & -2 & -1 Step 3: Calculate the inverse of A, denoted as A^-1. The formula for the inverse of a matrix is A^-1 = (1)/((A)) adj(A). Using (A) = 3 from part (a): A^-1 = (1)/(3) -3 & 3 & 0 \\ 1 & -2 & 2 \\ 4 & -2 & -1 A^-1 = (-3)/(3) & (3)/(3) & (0)/(3) \\ (1)/(3) & (-2)/(3) & (2)/(3) \\ (4)/(3) & (-2)/(3) & (-1)/(3) A^-1 = -1 & 1 & 0 \\ (1)/(3) & -(2)/(3) & (2)/(3) \\ (4)/(3) & -(2)/(3) & -(1)/(3) The inverse of A is -1 & 1 & 0 \\ (1)/(3) & -(2)/(3) & (2)/(3) \\ (4)/(3) & -(2)/(3) & -(1)/(3) .