Question

Let \( A \) be a \( 3 \times 3 \) matrix with \( |A| = 7 \). If \( B = 3A \), then the value of \[ \left| \frac{{adj } A}{B} \right| \] is equal to

• A. \( \frac{7}{3} \)
• B. \( \frac{7}{9} \)
• C. \( \frac{49}{9} \)
• D. \( \frac{7}{27} \)
• E. \( \frac{49}{27} \)

Hint:

For a scalar multiple \( B = kA \), the determinant scales as \( |B| = k^n |A| \).

Answer 1

The Correct Option is D

Step 1: Using the determinant property of adjugate matrix 
The determinant of adjugate matrix is related to the determinant of the original matrix by: \[ {adj } A = |A| A^{-1} \] The determinant property states: \[ |{adj } A| = |A|^{n-1} \] where \( n = 3 \) (since \( A \) is a \( 3 \times 3 \) matrix): \[ |{adj } A| = |A|^2 = 7^2 = 49 \] Step 2: Determinant of matrix \( B \) Since \( B = 3A \), we use: \[ |B| = 3^3 |A| = 27 \times 7 = 189 \] Step 3: Compute \( \left| \frac{{adj } A}{B} \right| \) \[ \left| \frac{{adj } A}{B} \right| = \frac{|{adj } A|}{|B|} \] \[ = \frac{49}{189} = \frac{7}{27} \] 
Final Answer: \[ \boxed{\frac{7}{27}} \]