Let A be a 2x2 matrix with real entries such that A^2 = 2A - I. Which of the following statements is/are necessarily true?
Step-by-Step Solution
Key Concept: A^2 - 2A + I = 0 => (A - I)^2 = 0.
Step 1: A^2 - 2A + I = 0 => (A - I)^2 = 0. Step 2: The minimal polynomial divides (x-1)^2, so the only eigenvalue is 1. Step 3: Trace = sum of eigenvalues = 2. Step 4: A is invertible since eigenvalues are non-zero. Step 5: However, A need not equal I; it can be a Jordan block [[1,1],[0,1]] which satisfies (A-I)^2 = 0 but A ≠ I. Thus A, C, D are true, B is false.
Correct Answer: A, C, D