Matrices & Determinants
Determinants
Grade Class 12
Question:
If α, β, γ satisfy the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>x</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>x</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math>, then
(A) α + β + γ = 0
(B) α<sup>2</sup> + β<sup>2</sup> + γ<sup>2</sup> = 6
(C) α<sup>3</sup> + β<sup>3</sup> + γ<sup>3</sup> = -6
(D) α<sup>4</sup> + β<sup>4</sup> + γ<sup>4</sup> = 18
Step-by-Step Solution
Key Concept: The determinant |x 1 1; 1 x 1; 1 1 x| = (x-1)^2(x+2). Setting this to 0 gives roots x=1, 1, -2. If alpha, beta, gamma are these roots, then alpha+beta+gamma = 1+1-2 = 0. Sum of squares = 1+1+4 = 6. Sum of cubes = 1+1-8 = -6. Sum of fourth powers = 1+1+16 = 18.
The determinant is (x-1)<sup>2</sup>(x+2) = 0. The roots are 1, 1, -2. Let α=1, β=1, γ=-2. Then α+β+γ = 1+1-2 = 0. α<sup>2</sup>+β<sup>2</sup>+γ<sup>2</sup> = 1+1+4 = 6. α<sup>3</sup>+β<sup>3</sup>+γ<sup>3</sup> = 1+1-8 = -6. α<sup>4</sup>+β<sup>4</sup>+γ<sup>4</sup> = 1+1+16 = 18.
Correct Answer: A,B,C,D