Consider the following statements in respect of two non-singular matrices A and B of the same order n:
1. adj(AB) = (adjA)(adjB)
2. adj(AB) = adj(BA)
3. (AB)adj(AB) − |AB|Iₙ is a null matrix of order n
How many of the above statements are correct?
A.None
B.Only one statement✓ Correct
C.Only two statements
D.All three statements
Explanation
Statement 1: adj(AB) = adj(B)·adj(A), not (adjA)(adjB). FALSE. Statement 2: adj(AB) = adj(B)adj(A) and adj(BA) = adj(A)adj(B); these are equal only when adj(A) and adj(B) commute — not generally true. FALSE. Statement 3: For any square matrix M, M·adj(M) = |M|·I, so (AB)adj(AB) − |AB|Iₙ = 0. TRUE. Only one statement is correct.
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