Question

Assertion (A): If \( B \) is a \( 3 \times 3 \) matrix and \( |B| = 6 \), then \( | {Adj}(B) | = 36 \). 
Reason (R): If \( B \) is a square matrix of order \( n \), then \( |{Adj}(B)| = |B|^n \).

• A. Both (A) and (R) are true and (R) is the correct explanation of (A)
• B. Both (A) and (R) are true but (R) is not the correct explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Hint:

For adjugate matrices, remember that \( |{Adj}(B)| = |B|^n \), where \( n \) is the order of the square matrix \( B \). This formula is key to solving determinant-related problems involving adjugates.

Answer 1

The Correct Option is A

We are given that \( B \) is a \( 3 \times 3 \) matrix and \( |B| = 6 \), and we are to determine if \( | {Adj}(B) | = 36 \). 
- From the reason (R), we know that for any square matrix \( B \) of order \( n \), the determinant of its adjugate matrix \( {Adj}(B) \) is given by: \[ |{Adj}(B)| = |B|^n. \] - For \( B \) being a \( 3 \times 3 \) matrix (\( n = 3 \)), we apply the formula: \[ |{Adj}(B)| = |B|^3 = 6^3 = 216. \] So, the assertion (A) that \( |{Adj}(B)| = 36 \) is false. Thus, (A) is false but (R) is true.