The determinant | cos (α +β ) - sin (α +β ) cos 2β sin α cos α sin β - cos α sin α cos β | is independent of

Question

The determinant | cos (α +β ) - sin (α +β ) cos 2β sin α cos α sin β - cos α sin α cos β | is independent of (A)  α  (B)  β  (C)  α and β  (D) Neither α nor β

Tardigrade Admin · Accepted Answer

We have the determinant | cos (α +β ) - sin (α +β ) cos 2β sin α cos α sin β - cos α sin α cos β | R1→ R1+(sinβ )R2+(cosβ )R3, | 0 0 1+ cos 2β sin α cos α sin β - cos α sin α cos β | =(1+ cos 2β ) ( sin 2α + cos 2α ) =1+ cos 2β which is independent of α .

Q. The determinant $∣ ∣ cos (α + β) sin α - cos α - sin (α + β) cos α sin α cos 2 β sin β cos β ∣ ∣$ is independent of

$α$

$β$

$α$ and $β$

Neither $α$ nor $β$

Q. The determinant ∣∣​cos(α+β)sinα−cosα​−sin(α+β)cosαsinα​cos2βsinβcosβ​∣∣​ is independent of

α

β

α and β

Neither α nor β

We have the determinant ∣∣​cos(α+β)sinα−cosα​−sin(α+β)cosαsinα​cos2βsinβcosβ​∣∣​ R1​→R1​+(sinβ)R2​+(cosβ)R3​, ∣∣​0sinα−cosα​0cosαsinα​1+cos2βsinβcosβ​∣∣​ =(1+cos2β)(sin2α+cos2α) =1+cos2β which is independent of α .

Q. The determinant $∣ ∣ cos (α + β) sin α - cos α - sin (α + β) cos α sin α cos 2 β sin β cos β ∣ ∣$ is independent of

$α$

$β$

$α$ and $β$

Neither $α$ nor $β$