Question

For a square matrix } A_{n \times n}:
(A) \( |\text{adj } A| = |A|^{n-1} \) 
(B) \( |A| = |\text{adj } A|^{n-1} \) 
(C) \( A (\text{adj } A) = |A| \) 
(D) \( |A^{-1}| = \frac{1}{|A|} \) 
$\text{Choose the \textbf{correct} answer from the options given below:}$

• A. (B) and (D) only
• B. (A) and (D) only
• C. (A), (C), and (C) only
• D. (B), (C), and (D) only

Hint:

When working with determinants, always ensure you're familiar with key properties like \( |\text{adj} \, A| = |A|^{n-1} \) and \( |A^{-1}| = \frac{1}{|A|} \). These properties are very helpful when simplifying matrix expressions and solving determinant-related problems. Be cautious with misstatements, such as \( |A| \neq |\text{adj} \, A|^{n-1} \), as they can lead to confusion.

Answer 1

The Correct Option is B

The determinant of the adjugate of a square matrix \( A_{n \times n} \) is \(|\text{adj} \, A| = |A|^{n-1}\).

This property validates option (A).

For the inverse of a matrix, the determinant is \(|A^{-1}| = \frac{1}{|A|}\).

This property validates option (D).

Option (C) is incorrect because the relation \( A (\text{adj} \, A) = |A| I \) is valid but not pertinent to the determinant properties being examined.

Option (B) is incorrect due to the misstatement \( |A| eq |\text{adj} \, A|^{n-1} \), which inverts the correct property.

Therefore, the correct options are (A) and (D).