Let A be a 3x3 matrix with real entries such that A^3 = I. Which of the following statements are necessarily true?
A. A is invertible.
B. The minimal polynomial of A divides x^3 - 1.
C. All eigenvalues of A are real.
D. The trace of A is an integer.
Step-by-Step Solution
Key Concept: Use properties of the minimal polynomial and roots of unity.
Step 1: Since A^3 = I, det(A)^3 = 1, so det(A) ≠ 0, hence A is invertible. Step 2: The minimal polynomial divides x^3 - 1 = (x-1)(x^2+x+1). Step 3: Eigenvalues are 1, ω, ω^2, not necessarily all real. Step 4: Trace is the sum of eigenvalues, each a root of unity; the sum is always an integer (possible traces: 3, 0, -3, 2, -1, etc.). Thus A, B, and D are true.
Correct Answer: A, B, D