Question

If \( A \) is a square matrix of order \(3\) and \( |A|=-3 \), then the value of \( |2AA^T| \) is:

• A. \( -36 \)
• B. \( 72 \)
• C. \( -72 \)
• D. \( 36 \)

Hint:

For determinant problems involving transpose and products, remember these key formulas: \[ |AB|=|A||B| \] \[ |A^T|=|A| \] \[ |AA^T|=|A|^2 \] Also, for an \(n\times n\) matrix: \[ |kA|=k^n|A| \] For a \(3\times3\) matrix specifically: \[ |2A|=2^3|A|=8|A| \]

Answer 1

The Correct Option is B

Concept: This problem is based on important determinant properties involving:
• Scalar multiplication of matrices,
• Product of matrices,
• Determinant of transpose matrices. The important formulas used are: \[ |AB|=|A||B| \] \[ |A^T|=|A| \] and for an \(n\times n\) matrix: \[ |kA|=k^n|A| \] where \(n\) is the order of the matrix. Since the matrix is of order \(3\), the scalar multiplication factor will be raised to power \(3\).

Step 1: Writing the given information We are given: \[ |A|=-3 \] We need to find: \[ |2AA^T| \] Since \(A\) is of order \(3\), the matrix \(AA^T\) is also of order \(3\).

Step 2: Using the scalar multiplication determinant property Using: \[ |kB|=k^n|B| \] for an \(n\times n\) matrix. Here: \[ n=3,\qquad k=2 \] Therefore, \[ |2AA^T| = 2^3|AA^T| \] \[ =8|AA^T| \]

Step 3: Evaluating \( |AA^T| \) Using the determinant property of product: \[ |AA^T|=|A||A^T| \] Also, determinant of transpose equals determinant of the original matrix: \[ |A^T|=|A| \] Hence, \[ |AA^T|=|A||A| \] \[ =|A|^2 \] Given: \[ |A|=-3 \] Therefore, \[ |AA^T|=(-3)^2 \] \[ =9 \]

Step 4: Substituting into the required expression We found: \[ |2AA^T|=8|AA^T| \] and \[ |AA^T|=9 \] Therefore, \[ |2AA^T|=8\times 9 \] \[ =72 \] Final Answer: \[ \boxed{72} \] Therefore, the correct option is: \[ \boxed{(B)} \]