For a square matrix $A_{n \times n}$
(A) $|adj A|=|A|^{n-1}$
(B) $|A|=|adj A|^{n-1}$
(C) $A(adj A)=|A|$
(D) $\left|A^{-1}\right|=\frac{1}{|A|}$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option 2: (A) and (D) only
$\text{(A)}\ |\text{adj}A| = |A|^{n-1}\ \text{(True)}$
$\text{(B)}\ |A| = |\text{adj}A|^{\,n-1} = (|A|^{n-1})^{n-1} = |A|^{(n-1)^2}\ \text{(False)}$
$\text{(C)}\ A(\text{adj}A)=|A|I\ \text{(not } |A|)\ \text{(False)}$
$\text{(D)}\ |A^{-1}|=\frac{1}{|A|}\ \text{(True)}$
$\text{Correct options: A and D}$