If A and B are two invertible matrices, then which of the following statements are correct?
(A) $|A^{-1}| = |A|^{-1}$
(B) $adj\,A = |A|A^{-1}$
(C) $(AB)^{-1}=A^{-1}B-1$
(D) $(A + B)^{-1}=A^{-1}+B^{-1}$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
$\textbf{Check each statement carefully}$
$\textbf{(A)}\;|A^{-1}|=|A|^{-1}$ is true because for any invertible matrix, determinant of inverse is reciprocal of determinant.
$\textbf{(B)}\;\text{adj}\,A=|A|\;A^{-1}$ is true (standard identity for invertible matrices).
$\textbf{(C)}\;(AB)^{-1}=A^{-1}B^{-1}$ is false because correct formula is
$\qquad (AB)^{-1}=B^{-1}A^{-1}$
$\textbf{(D)}\;(A+B)^{-1}=A^{-1}+B^{-1}$ is false (this is never generally true).
Thus the correct statements are: (A) and (B).