Matrices & Determinants
Determinants
Grade Class 12
Question:
The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>cos</mi><mo>(</mo><mi>θ</mi><mo>+</mo><mi>ϕ</mi><mo>)</mo></mtd><mtd><mo>-</mo><mi>sin</mi><mo>(</mo><mi>θ</mi><mo>+</mo><mi>ϕ</mi><mo>)</mo></mtd><mtd><mi>cos</mi><mn>2</mn><mi>ϕ</mi></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>ϕ</mi></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>ϕ</mi></mtd></mtr></mtable></mfenced></math> is -
(A) 0
(B) independent of θ
(C) independent of ϕ
(D) independent of θ & ϕ both
Step-by-Step Solution
Key Concept: Expand the determinant or use row/column operations to simplify the expression. Notice that the first row can be expressed in terms of the other rows using trigonometric addition formulas.
Let the determinant be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>cos</mi><mi>θ</mi><mi>cos</mi><mi>ϕ</mi><mo>-</mo><mi>sin</mi><mi>θ</mi><mi>sin</mi><mi>ϕ</mi></mtd><mtd><mo>-</mo><mo>(</mo><mi>sin</mi><mi>θ</mi><mi>cos</mi><mi>ϕ</mi><mo>+</mo><mi>cos</mi><mi>θ</mi><mi>sin</mi><mi>ϕ</mi><mo>)</mo></mtd><mtd><mi>cos</mi><mn>2</mn><mi>ϕ</mi></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>ϕ</mi></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>ϕ</mi></mtd></mtr></mtable></mfenced></math>. Performing <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub><mo>→</mo><msub><mi>R</mi><mn>1</mn></msub><mo>-</mo><mi>cos</mi><mi>ϕ</mi><mo>(</mo><mi>cos</mi><mi>θ</mi><msub><mi>R</mi><mn>2</mn></msub><mo>-</mo><mi>sin</mi><mi>θ</mi><msub><mi>R</mi><mn>3</mn></msub><mo>)</mo><mo>+</mo><mi>sin</mi><mi>ϕ</mi><mo>(</mo><mi>sin</mi><mi>θ</mi><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><mi>cos</mi><mi>θ</mi><msub><mi>R</mi><mn>3</mn></msub><mo>)</mo></math> or simply expanding shows the result is independent of θ.
Correct Answer: B