Matrices & Determinants
Determinant of 3x3
Grade Class 12
Question:
There are two numbers <i>x</i> making the value of the determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mi>x</mi></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>2</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math> equal to 86. The sum of these two numbers, is-
(A) -4
(B) 5
(C) -3
(D) 9
Step-by-Step Solution
Key Concept: Expand the determinant along the first column or row to form a quadratic equation in x, then use the sum of roots formula (-b/a).
Expanding the determinant along the first column: 1(x(2x) - (-1)(4)) - 2((-2)(2x) - 5(4)) + 0 = 86. This simplifies to 1(2x^2 + 4) - 2(-4x - 20) = 86, which is 2x^2 + 4 + 8x + 40 = 86. Rearranging gives 2x^2 + 8x - 42 = 0, or x^2 + 4x - 21 = 0. The sum of the roots is -4.
Correct Answer: 1