Question

If and I is the identity matrix of order 2, show that

Answer 1

The given matrix is,

A=[ 0 −tan α 2 tan α 2 0 ]

Calculate the left hand side of the equation I+A=( I−A )[ cosα −sinα sinα cosα ].

I+A=[ 1 0 0 1 ]+[ 0 −tan α 2 tan α 2 0 ] =[ 1 −tan α 2 tan α 2 1 ]

Calculate right hand side of the equation I+A=( I−A )[ cosα −sinα sinα cosα ].

( I−A )[ cosα −sinα sinα cosα ]=( [ 1 0 0 1 ]−[ 0 −tan α 2 tan α 2 1 ] )[ cosα −sinα sinα cosα ] =[ 1 tan α 2 −tan α 2 1 ][ cosα −sinα sinα cosα ] =[ cosα+sinαtan α 2 −sinα+cosαtan α 2 −cosαtan α 2 +sinα sinαtan α 2 +cosα ]

Simplify further,

( I−A )[ cosα −sinα sinα cosα ] =[ 1−2 sin 2 α 2 +2sin α 2 cos α 2 tan α 2 −2sin α 2 cos α 2 +( 2 cos 2 α 2 −1 )tan α 2 −( 2 cos 2 α 2 −1 )tan α 2 +2sin α 2 cos α 2 2 sin 2 α 2 +1−2 sin 2 α 2 ] =[ 1 −tan α 2 tan α 2 1 ]

Hence, it is proved that L.H.S.=R.H.S..