Question

Find the value of \(\rm \begin{vmatrix} 0 & c & b\\ c & 0 &a \\ b& a & 0 \end{vmatrix}^2\)
1. a²b²c²
2. 4a2b2c2
3. 2a2b2c2
4. (a + b + c)²

Answer 1

Correct Answer - Option 2 : 4a2b2c2

Concept:

To evaluate any 3 × 3 determinant expansion by minors can be used. This method involves multiplying the elements in the first column of the determinant by the cofactors of those elements. Then subtract the middle product and add the final product. For a 3 × 3 determinant, we can write,

\(\rm \begin{vmatrix} a & b & c\\ x & y &z \\ p& q & r \end{vmatrix} = a\begin{vmatrix} y& z\\ q &r \end{vmatrix} - b\begin{vmatrix} x &z \\ p & r \end{vmatrix} +c\begin{vmatrix} x &y \\ p & q \end{vmatrix}\)

The expansion can be done based on any row or column.

Calculation:

Given determinant is \(\rm \begin{vmatrix} 0 & c & b\\ c & 0 &a \\ b& a & 0 \end{vmatrix}^2\), simplifying the determinant based on first row we get,

= {0 × (0 - a²) - c × (c× 0 - b × a) + b(c × a - b × 0)}²

= (abc + abc)²

= 4a2b2c2

So, the right answer is 4a2b2c2