Question

Given a matrix \(A\) of order \(3\times3\). If \[ |A|=3 \] then the value of \[ |A(\operatorname{adj}A)| \] is:

• A. \(3\)
• B. \(27\)
• C. \(9\)
• D. \(81\)

Hint:

For an \(n\times n\) matrix: \[ |\operatorname{adj}A|=|A|^{n-1} \]

Answer 1

The Correct Option is B

Step 1: Use the property of determinant For any square matrix: \[ |AB|=|A||B| \] Hence: \[ |A(\operatorname{adj}A)| = |A|\cdot|\operatorname{adj}A| \]
Step 2: Use determinant property of adjoint matrix For an \(n\times n\) matrix: \[ |\operatorname{adj}A|=|A|^{\,n-1} \] Since matrix \(A\) is of order: \[ 3\times3 \] we have: \[ |\operatorname{adj}A|=|A|^2 \] Given: \[ |A|=3 \] Thus: \[ |\operatorname{adj}A|=3^2=9 \]
Step 3: Find required value \[ |A(\operatorname{adj}A)| = |A|\cdot|\operatorname{adj}A| \] \[ =3\times9 \] \[ =27 \] Option analysis:
• Option (A): Incorrect
• Option (B): Correct
• Option (C): Incorrect
• Option (D): Incorrect Hence: \[ \boxed{\text{(B)}} \]