If Aα=[ cos α sin α - sin α cos α], then Aα Aβ is equal to

Question

If Aα=[ cos α sin α - sin α cos α], then Aα Aβ is equal to (A) Aα+β (B) Aα β (C) Aα-β (D) none of these

Tardigrade Admin · Accepted Answer

We have Aα Aβ =[ cos α sin α - sin α cos α ][ cos β sin β - sin β cos β ] =[ cos α cos β- sin α sin β cos α sin β+ sin α cos β - sin α cos β- cos α sin β - sin α sin β+ cos α cos β ] =[ cos (α+β) sin (α+β) - sin (α+β) cos (α+β) ]=Aα+β

Q. If $A_{α} = [cos α - sin α sin α cos α]$ , then $A_{α} A_{β}$ is equal to

$A_{α + β}$

$A_{α β}$

$A_{α - β}$

none of these

We have
$A_{α} A_{β} = [cos α - sin α sin α cos α] [cos β - sin β sin β cos β]$
$= [cos α cos β - sin α sin β cos α sin β + sin α cos β - sin α cos β - cos α sin β - sin α sin β + cos α cos β]$
$= [cos (α + β) - sin (α + β) sin (α + β) cos (α + β)] = A_{α + β}$

Q. If Aα​=[cosα−sinα​sinαcosα​], then Aα​Aβ​ is equal to

Aα+β​

Aαβ​

Aα−β​