Understanding the Problem: Finding |adj A|

The question asks us to find the value of the determinant of the adjoint matrix, denoted as $|adj A|$, for a given 3x3 matrix $A$.

The matrix $A$ is given as:

$A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$

Key Concept: Determinant of the Adjoint Matrix

There's a useful property connecting the determinant of a matrix $A$ and the determinant of its adjoint matrix $adj A$. For an $n \times n$ matrix $A$, this property is:

$|adj A| = |A|^{n-1}$

To find $|adj A|$, we first need to calculate the determinant of matrix $A$ ($|A|$) and know the order of the matrix ($n$).

Step 1: Calculate the Determinant of Matrix A (|A|)

The matrix $A$ is a 3x3 matrix. We can calculate its determinant using the standard formula. Let's expand along the first row:

$|A| = 0 \cdot \begin{vmatrix} \sqrt{3} & 0 \\ 0 & 0 \end{vmatrix} - 0 \cdot \begin{vmatrix} 0 & 0 \\ \sqrt{3} & 0 \end{vmatrix} + \sqrt{3} \cdot \begin{vmatrix} 0 & \sqrt{3} \\ \sqrt{3} & 0 \end{vmatrix}$

Now, calculate the determinants of the 2x2 matrices:

• $\begin{vmatrix} \sqrt{3} & 0 \\ 0 & 0 \end{vmatrix} = (\sqrt{3} \times 0) - (0 \times 0) = 0 - 0 = 0$
• $\begin{vmatrix} 0 & 0 \\ \sqrt{3} & 0 \end{vmatrix} = (0 \times 0) - (0 \times \sqrt{3}) = 0 - 0 = 0$
• $\begin{vmatrix} 0 & \sqrt{3} \\ \sqrt{3} & 0 \end{vmatrix} = (0 \times 0) - (\sqrt{3} \times \sqrt{3}) = 0 - 3 = -3$

Substitute these values back into the determinant calculation for $A$:

$|A| = 0 \cdot (0) - 0 \cdot (0) + \sqrt{3} \cdot (-3)$ $|A| = 0 - 0 - 3\sqrt{3}$ $|A| = -3\sqrt{3}$

Step 2: Calculate |adj A| using the Property

We know that $n = 3$ for matrix $A$. Using the formula $|adj A| = |A|^{n-1}$:

$|adj A| = |A|^{3-1}$ $|adj A| = |A|^2$

Substitute the value of $|A|$ we found:

$|adj A| = (-3\sqrt{3})^2$

To calculate this, we square both the coefficient and the square root term:

$|adj A| = (-3)^2 \times (\sqrt{3})^2$ $|adj A| = 9 \times 3$ $|adj A| = 27$

Conclusion

Therefore, the value of $|adj A|$ for the given matrix $A$ is 27.