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

Answer 1

Answer: (B)

(A) and (D) only

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.

(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 $(D)$.