Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $a, b, c$ are non-zero real numbers such that $$\begin{vmatrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \end{vmatrix} = 0$$, then:
\frac{1}{a} + \frac{1}{b} + \frac{1}{c^2} = 0
\frac{1}{a} + \frac{1}{b^2} + \frac{1}{c} = 0
\frac{1}{a^2} + \frac{1}{b} + \frac{1}{c} = 0
None of these
Step-by-Step Solution
Key Concept: Cyclic determinants factor using the sum of cubes identity and cube roots of unity.
For the determinant $\begin{vmatrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \end{vmatrix} = 0$, we use the identity $(ab)^3 + (bc)^3 + (ca)^3 - 3(ab)(bc)(ca) = 0$ which factors as $(ab + bc + ca)^3 - 3(ab)(bc)(ca) = 0$. This can be rewritten as three separate equations: $ab + bcw^2 + caw = 0$, $abw + bc + ca^2 = 0$, and $abw^2 + bcw + ca = 0$, where $w$ is a cube root of unity. Dividing by $abc$ gives $\frac{1}{cw^2} + \frac{1}{a} + \frac{1}{bw} = 0$ and cyclic permutations.
Correct Answer: 1,2,3