Question

If , show that .

Answer 1

The given matrix is,

F( x )=[ cosx −sinx 0 sinx cosx 0 0 0 1 ]

We have to prove that F( x )F( y )=F( x+y ).

The above matrix can also be written as,

F( y )=[ cosy −siny 0 siny cosy 0 0 0 1 ]

And,

F( x+y )=[ cos( x+y ) −sin( x+y ) 0 sin( x+y ) cos( x+y ) 0 0 0 1 ]

Also,

F( x )F( y )=[ cosx −sinx 0 sinx cosx 0 0 0 1 ][ cosy −siny 0 siny cosy 0 0 0 1 ] =[ cosxcosy−sinxsiny+0 −cosxsiny−sinxcosy+0 0 sinxcosy+cosxsiny+0 −sinxsiny+cosxcosy+0 0 0 0 1 ] =[ cos( x+y ) −sin( x+y ) 0 sin( x+y ) cos( x+y ) 0 0 0 1 ] =F( x+y )

Hence, it is proved that F( x )F( y )=F( x+y ).