If an idempotent matrix is also skew symmetric then it must be-
Step-by-Step Solution
Key Concept: An idempotent matrix satisfies A^2 = A. A skew-symmetric matrix satisfies A^T = -A. If A is both, then A^2 = A implies A^T A = A^T A^T = (-A)(-A) = A^2 = A. Also, A^T A = (-A)A = -A^2 = -A. Thus A = -A, which implies 2A = 0, so A = 0.
Given A is idempotent, A^2 = A. Given A is skew-symmetric, A^T = -A. Since A^2 = A, we have A^T A = (-A)A = -A^2 = -A. Also, for any matrix, A^T A is related to the norm. Specifically, if A^2 = A and A^T = -A, then A = A^2 = A A^T A = A (-A^T) = -A A^T. Actually, a simpler way: A^2 = A. Taking transpose, (A^T)^2 = A^T. Since A^T = -A, (-A)^2 = -A, so A^2 = -A. Since A^2 = A, we have A = -A, which implies 2A = 0, so A = 0 (null matrix).
Correct Answer: 4