The given matrix is,
A=[ 1 0 0 0 cosα sinα 0 sinα −cosα ]
The inverse of a matrix exists only if it satisfies the condition of | A |≠0.
So, the determinant of A is,
| A |=1( − cos 2 α− sin 2 α ) =−1
Since, | A |≠0, so inverse of the matrix A exists.
The formula to calculate the inverse of the matrix is,
A −1 = adjA | A | (1)
The cofactors of each element of the matrix is,
A 11 = ( −1 ) 1+1 [ − cos 2 α− sin 2 α ] =−1
A 12 = ( −1 ) 1+2 [ 0( −cosα )−0sinα ] =0
A 13 = ( −1 ) 1+3 [ ( 0 )sinα−( 0 )cosα ] =0
A 21 = ( −1 ) 2+1 [ 0( −cosα )−( 0 )sinα ] =0
A 22 = ( −1 ) 2+2 [ −cosα−0 ] =−cosα
A 23 = ( −1 ) 2+3 ( sinα−0 ) =−sinα
A 31 = ( −1 ) 3+1 [ 0−0 ] =0
A 32 = ( −1 ) 3+2 [ 1×sinα−0×0 ] =−sinα
A 33 = ( −1 ) 3+3 [ cosα−0 ] =cosα
The adjoint of A will be,
adjA=[ A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 ] =[ −1 0 0 0 −cosα −sinα 0 −sinα cosα ]
Substitute [ −1 0 0 0 −cosα −sinα 0 −sinα cosα ] for adjA and −1 for | A | in equation (1),
A −1 =−1[ −1 0 0 0 −cosα −sinα 0 −sinα cosα ] =[ 1 0 0 0 cosα sinα 0 sinα −cosα ]