Question

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

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

Answer 2

Assertion (A): If \( B \) is a \( 3 \times 3 \) matrix and \( |B| = 6 \), then \[ |\text{Adj}(B)| = 36. \]

Reason (R): If \( B \) is a square matrix of order \( n \), then \[ |\text{Adj}(B)| = |B|^{n-1}. \]

Explanation: For a square matrix \( B \) of order \( n \), the determinant of its adjoint is related to the determinant of \( B \) by the formula: \[ |\text{Adj}(B)| = |B|^{n-1}. \] Since \( B \) is \( 3 \times 3 \), \( n = 3 \), so \[ |\text{Adj}(B)| = |B|^{2} = 6^2 = 36. \] Therefore, both (A) and (R) are true, and (R) correctly explains (A).

Final answer: Both (A) and (R) are true and (R) is the correct explanation of (A).