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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Here are the solutions to the questions: 1. Matrix A Given matrix A = 2x & 2 \\ 3 & x . i) To find the positive value of x for which the determinant of A is 12: The determinant of a 2 × 2 matrix a & b \\ c & d is ad - bc. For matrix A, the determinant is: (A) = (2x)(x) - (2)(3) = 2x^2 - 6 We are given that (A) = 12: 2x^2 - 6 = 12 2x^2 = 12 + 6 2x^2 = 18 x^2 = (18)/(2) x^2 = 9 x = ±sqrt(9) x = ± 3 Since we need the positive value of x: x = 3 ii) To write A^-1: First, substitute x=3 into matrix A: A = 2(3) & 2 \\ 3 & 3 = 6 & 2 \\ 3 & 3 The determinant of A is 12 (as found in part (i)). The inverse of a 2 × 2 matrix a & b \\ c & d is (1)/(ad-bc) d & -b \\ -c & a . Using this formula for matrix A: A^-1 = (1)/(12) 3 & -2 \\ -3 & 6 A^-1 = (3)/(12) & -(2)/(12) \\ -(3)/(12) & (6)/(12) = (1)/(4) & -(1)/(6) \\ -(1)/(4) & (1)/(2) 2. Solve the equation x^2 - 4x - 2 = 0, giving your answers correct to 2 decimal places. This is a quadratic equation in the form $ax^2 + bx +