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|I_{n} is a null matrix of order n How many of the above statements are correct?
- A. None
- B. Only one statement ✓
- C. Only two statements
- D. All three statements
Correct Answer: B. Only one statement
Explanation
Statement 1 is false because the reversal law applies: adj(AB) = adj(B)adj(A). Statement 2 is false as matrix multiplication is not commutative, so adj(AB) \neq adj(BA) in general. Statement 3 is true because for any non-singular matrix M, M \cdot adj(M) = |M|I. Substituting M = AB yields (AB)adj(AB) = |AB|I_n, which makes the expression a null matrix. Thus, <strong>ONLY</strong> one statement is correct.
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