Matrices & Determinants
Determinant Properties
Grade Class 12
Question:
The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup></mtd><mtd><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mi>z</mi></mrow></mfenced><mn>2</mn></msup></mtd><mtd><mi>y</mi><mi>z</mi></mtd></mtr><mtr><mtd><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd><msup><mfenced><mrow><mi>z</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mtd><mtd><mi>z</mi><mi>x</mi></mtd></mtr><mtr><mtd><msup><mi>z</mi><mn>2</mn></msup></mtd><mtd><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup></mtd><mtd><mi>x</mi><mi>y</mi></mtd></mtr></mtable></mfenced></math> is divisible by -
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi></math>
(C) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mi>z</mi></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></math>
Step-by-Step Solution
Key Concept: The determinant can be simplified by performing row operations and factoring out common terms, revealing factors like (x-y), (y-z), (z-x), and (x+y+z).
Let the determinant be \Delta. Performing operations R1 -> R1 - R2 and R2 -> R2 - R3, we get factors (x-y) and (y-z). Further simplification leads to the factor (x+y+z) and (x^2+y^2+z^2).
Correct Answer: A,B,D