Question

If A = $\begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$, then |adj A| is equal to

• A. 3
• B. 9
• C. 27
• D. 81

Hint:

For determinants of matrices with many zeros, always expand along the row or column containing the most zeros to simplify the calculation. Also, remember the key property $|\text{adj}(A)| = |A|^{n-1}$, which is frequently tested.

Answer 1

The Correct Option is C

Step 1: Concept Identification: This question assesses the relationship between the determinant of a matrix and the determinant of its adjugate.
Step 2: Relevant Formula: For any n x n matrix A, the determinant of its adjugate is given by \(|\text{adj}(A)| = |A|^{n-1}\).
Step 3: Calculation: Given a 3x3 matrix, n=3. Thus, \(|\text{adj}(A)| = |A|^{3-1} = |A|^2\).
First, calculate |A|:
\[ |A| = \begin{vmatrix} 0 & 0 & \sqrt{3}
0 & \sqrt{3} & 0
\sqrt{3} & 0 & 0 \end{vmatrix} \]
Expanding along the first row:
\[ |A| = 0(\dots) - 0(\dots) + \sqrt{3}((0)(0) - (\sqrt{3})(\sqrt{3})) \]
\[ |A| = \sqrt{3} (0 - 3) = -3\sqrt{3} \]
Now, apply the formula for \(|\text{adj}(A)|\):
\[ |\text{adj}(A)| = |A|^2 = (-3\sqrt{3})^2 \]
\[ |\text{adj}(A)| = (-3)^2 \cdot (\sqrt{3})^2 = 9 \cdot 3 = 27 \]
Step 4: Conclusion: The determinant of the adjugate of matrix A is 27.