Let A be any square matrix of order n, then which of the following are true?
(A) $|adj\, A| = |A|^{n-1}$
(B) $|A^{-1}|=\frac{1}{|A|}$
(C) $|adj\, A| = |A|^n$
(D) $(A^T)^{-1}= (A^{-1})^T$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
(A) $|\text{adj }A| = |A|^{\,n-1}$
This is a standard identity for any square matrix of order $n$.
(A) is true.
(B) $|A^{-1}| = \frac{1}{|A|}$
Determinant of inverse is reciprocal of determinant.
(B) is true.
(C) $|\text{adj }A| = |A|^{\,n}$
This is wrong: the correct formula is $|\text{adj }A| = |A|^{\,n-1}$.
(C) is false.
(D) $(A^{T})^{-1} = (A^{-1})^{\,T}$
Inverse and transpose commute for any invertible matrix.
(D) is true.
The true statements are (A), (B), and (D).