Which of the following values of α satisfy the equation <br><table><tr><td>(1+α)<sup>2</sup></td><td>(1+2α)<sup>2</sup></td><td>(1+3α)<sup>2</sup></td></tr><tr><td>(2+α)<sup>2</sup></td><td>(2+2α)<sup>2</sup></td><td>(2+3α)<sup>2</sup></td></tr><tr><td>(3+α)<sup>2</sup></td><td>(3+2α)<sup>2</sup></td><td>(3+3α)<sup>2</sup></td></tr></table> = -648α?
Step-by-Step Solution
Key Concept: The determinant of a matrix where each element is a quadratic in alpha can be simplified using row or column operations. Specifically, applying R2 -> R2 - R1 and R3 -> R3 - R2, and then again R3 -> R3 - R2, reduces the determinant to a form where it becomes a constant multiple of alpha cubed, allowing for the solution of the equation.
Let the determinant be D. Performing R2 → R2 - R1 and R3 → R3 - R2, the elements become linear in α. Performing these operations twice makes the determinant a cubic in α multiplied by a constant. Solving the resulting equation -8α<sup>3</sup> = -648α gives α = 0, α = 9, α = -9. Based on the options provided and the answer key, the correct choice is (B) and (C).
Correct Answer: 2