Question

If \(A\) is a \(3\times3\) matrix with \[ |A|=-1, \] then \[ \left|\,3(\operatorname{adj}(A^T))A^2\,\right| \] equals

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

Hint:

For a \(3\times3\) matrix: \[ |\operatorname{adj}(A)|=|A|^2 \] and \[ |kA|=k^3|A| \] These two formulas are frequently used in determinant problems.

Answer 1

The Correct Option is C

Concept: For an \(n\times n\) matrix: \[ |kA|=k^n|A| \] \[ |\operatorname{adj}(A)| = |A|^{\,n-1} \] and \[ |A^m|=|A|^m \]

Step 1: Evaluate the determinant of each factor. Since \(A\) is \(3\times3\), \[ |3B|=3^3|B| \] where \[ B=(\operatorname{adj}(A^T))A^2 \] Thus \[\begin{aligned} \left|3(\operatorname{adj}(A^T))A^2\right| = 3^3 \left|\operatorname{adj}(A^T)\right| |A^2| \end{aligned}\]

Step 2: Compute the determinant of the adjoint. \[ |A^T|=|A|=-1 \] For a \(3\times3\) matrix, \[ \left|\operatorname{adj}(A^T)\right| = |A^T|^{2} = (-1)^2 = 1 \]

Step 3: Compute \(|A^2|\). \[ |A^2| = |A|^2 = (-1)^2 = 1 \]

Step 4: Find the required determinant. \[\begin{aligned} 3^3\times1\times1 &=27 \end{aligned}\] \[\begin{aligned} \boxed{27} \end{aligned}\] Hence, option \(\mathbf{(C)}\) is correct.