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: Identify the given matrix X and the identity matrix I. The given matrix is: X = -2 & 1 \\ 2 & 4 For a 2 × 2 matrix, the identity matrix I is: I = 1 & 0 \\ 0 & 1 Step 2: Calculate I. Multiply the identity matrix by the scalar : I = 1 & 0 \\ 0 & 1 = & 0 \\ 0 & Step 3: Calculate X - I. Subtract I from X: X - I = -2 & 1 \\ 2 & 4 - & 0 \\ 0 & = -2 - & 1 - 0 \\ 2 - 0 & 4 - = -2 - & 1 \\ 2 & 4 - Step 4: Set the determinant |X - I| to zero. The determinant of a 2 × 2 matrix a & b \\ c & d is ad - bc. So, for X - I: |X - I| = (-2 - )(4 - ) - (1)(2) We are given that |X - I| = 0: (-2 - )(4 - ) - 2 = 0 Step 5: Expand and simplify the equation to find . Expand the product: -8 + 2 - 4 + ^2 - 2 = 0 Combine like terms: ^2 - 2 - 10 = 0 Step 6: Solve the quadratic equation for using the quadratic formula. The quadratic formula for a^2 + b + c = 0 is = -b ± sqrt(b^2 - 4ac)2a. Here, a=1, b=-2, c=-10. = -(-2) ± sqrt((-2)^2 - 4(1)(-10))2(1) = 2 ± sqrt(4 + 40)2 = 2 ± sqrt(44)2 Simplify sqrt(44): sqrt(44) = sqrt(4 × 11) = sqrt(4) × sqrt(11) = 2sqrt(11) Substitute this back into the expression for : = 2 ± 2sqrt(11)2 Factor out 2 from the numerator: = 2(1 ± sqrt(11))2 = 1 ± sqrt(11) The values of are 1 + sqrt(11) and 1 - sqrt(11). The final answer is 1 ± sqrt(11).