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 \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

• A. 2
• B. 3
• C. 0
• D. 1

Answer 1

The Correct Option is C

To resolve the problem, we will calculate the determinant and examine its implications.

The determinant is provided as:

\[\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0\]

This equation signifies that the determinant of the presented 3x3 matrix is zero. A determinant of zero indicates that the rows (or columns) of the matrix are linearly dependent.

Applying the condition of a zero determinant yields:

\[\alpha (\beta \gamma - c^2) - b(a \gamma - ac) + c(a b - a \beta) = 0\]

Upon simplification, a relationship between \(\alpha, \beta, \gamma, a, b,\) and \(c\) is established that results in this determinant equaling zero.

The objective is to compute:

\[\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c}\]

Given the linear dependency established by the zero determinant, the denominators \((\alpha - a), (\beta - b),\) and \((\gamma - c)\) are constrained such that their collective impact nullifies the sum.

This occurs when the conditions of the determinant and the interrelationships between \(a, \alpha, b, \beta, \gamma, c\) lead to the summation resolving to zero, potentially due to symmetry, an identity, or an unstated consequence of the dependency.

Conclusion: The expression evaluates to 0.

The definitive answer is 0, based on the assumption that the linear dependency forces the terms within the expression to interact and cancel each other out.