Matrices & Determinants
Determinant Zero — Derived Relation
nta_pyq_2024_apr
Grade 12
Question:
If $\alpha\neq a$, $\beta\neq b$, $\gamma\neq c$ and $\begin{vmatrix}\alpha&b&c\\a&\beta&c\\a&b&\gamma\end{vmatrix}=0$, then $\dfrac{a}{\alpha-a}+\dfrac{b}{\beta-b}+\dfrac{\gamma}{\gamma-c}$ is equal to:
Step-by-Step Solution
Key Concept: Apply $R_1\to R_1-R_2$, $R_2\to R_2-R_3$, then expand the determinant. After simplification, derive a relation among $\frac{a}{\alpha-a}$, $\frac{b}{\beta-b}$, $\frac{\gamma}{\gamma-c}$.
Row operations give: $\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}=2$.
Correct Answer: 4