Question

Find the value of \( \left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \end{matrix} \right|^2 \)

• A. \( a^2b^2c^2 \)
• B. \( 4a^2b^2c^2 \)
• C. \( 2a^2b^2c^2 \)
• D. \( (a+b+c)^2 \)

Hint:

For any skew-symmetric-like structure with zeros on the diagonal and symmetric off-diagonal terms, the determinant often simplifies to \( 2abc \). Memorizing this standard form helps in solving such determinant questions instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to evaluate the square of the determinant of a given \( 3 \times 3 \) symmetric matrix.

Step 2: Key Formula or Approach:
First, calculate the determinant of the matrix \( D \), then square it.
The determinant of a \( 3 \times 3 \) matrix is evaluated by expanding along any row or column.

Step 3: Detailed Explanation:
Let the determinant be:
\[ D = \left| \begin{matrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \end{matrix} \right| \] Expanding along the first row:
\[ D = 0 \cdot (0 \cdot 0 - a \cdot a) - c \cdot (c \cdot 0 - a \cdot b) + b \cdot (c \cdot a - 0 \cdot b) \] \[ D = 0 - c \cdot (-ab) + b \cdot (ca) \] \[ D = abc + abc = 2abc \] Now, squaring the determinant:
\[ D^2 = (2abc)^2 = 4a^2b^2c^2 \] This matches option (B).

Step 4: Final Answer:
(B) \( 4a^2b^2c^2 \)