Consider the following statements in respect of a non-singular matrix A of order n:
1. A(adjA^T) = A(adjA)^T
2. If A² = A, then A is identity matrix of order n
3. If A³ = A, then A is identity matrix of order n
Which of the statements given above are correct?
A.1 and 2 only✓ Correct
B.2 and 3 only
C.1 and 3 only
D.1, 2 and 3
Explanation
Statement 1: For non-singular A, adj(A^T) = (adj A)^T, so A·adj(A^T) = A·(adj A)^T. TRUE. Statement 2: If A is non-singular and A² = A, then multiplying by A⁻¹ gives A = I. TRUE. Statement 3: If A is non-singular and A³ = A, then A² = I, which means A could be I or −I (or other matrices with A² = I). Not necessarily identity. FALSE. So 1 and 2 only.
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