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

For a square matrix \( A_{n \times n} \), the determinant of the adjugate of \( A \) is given by:

\[|\text{adj} \, A| = |A|^{n-1}.\]

This property confirms that (A) is correct.

For the inverse of a matrix:

\[|A^{-1}| = \frac{1}{|A|}.\]

This property confirms that (D) is correct.

(C) is not part of the correct answer because while the relation \( A (\text{adj} \, A) = |A| I \) is valid, it is not relevant to the determinant properties discussed here.

(B) is incorrect because \( |A| \neq |\text{adj} \, A|^{n-1} \). It is a misstatement of the property.

Thus, the correct options are:

\((A)\) and \((D)\).

Answer 2

For a square matrix \( A_{n \times n} \), the determinant of the adjugate of \( A \) is given by:

\[ |\text{adj} \, A| = |A|^{n-1}. \]

This property confirms that (A) is correct.

For the inverse of a matrix:

\[ |A^{-1}| = \frac{1}{|A|}. \]

This property confirms that (D) is correct.

Explanation for (C):

While the relation \( A (\text{adj} \, A) = |A| I \) is valid, it is not relevant to the determinant properties discussed here. Hence, (C) is not part of the correct answer.

Explanation for (B):

(B) is incorrect because \( |A| \neq |\text{adj} \, A|^{n-1} \). It is a misstatement of the property.

Conclusion:

The correct options are:

• (A)
• (D)