Let A be a 3x3 matrix such that A^2 = A. If det(A) = 0 and the trace of A is 2, then the rank of A is:
Step-by-Step Solution
Key Concept: For an idempotent matrix A (A^2 = A), the eigenvalues are either 0 or 1. The trace of A is the sum of its eigenvalues, and the rank of A is the number of non-zero eigenvalues.
Since A^2 = A, the eigenvalues of A are 0 or 1. Let the eigenvalues be \lambda1, \lambda2, \lambda3. Given trace(A) = \lambda1 + \lambda2 + \lambda3 = 2. Since det(A) = 0, at least one eigenvalue must be 0. Thus, the eigenvalues are 1, 1, 0. The rank of an idempotent matrix is equal to the number of non-zero eigenvalues, which is 2.
Correct Answer: 2