If B=
( b 0 0 )
( 0 b 0 ) = bI, where I is the identity matrix,
( 0 0 b )
then B^7=
( b^7 0 0 )
( 0 b^7 0 ) =
( 0 0 b^7 )
So
( b^7 0 0 ) ( 3b 0 0 ) ( 1 0 0 ) ( 0 0 0 )
( 0 b^7 0 ) - ( 0 3b 0 ) + ( 0 1 0 ) = ( 0 0 0 ).
( 0 0 b^7 ) ( 0 0 3b ) ( 0 0 1 ) ( 0 0 0 )
b^7-3b+1=0 applies right across the matrices and this equation has 3 real roots which we can call b1, b2, b3, where b1, b2, b3 =-1.249, 0.333, 1.133 approx.
B=bI so the inverse of B=B'=I÷b, where b can be any one of b1, b2, b3.
B'=
( 1/b 0 0 ) ( -0.801 0 0 ) ( 3.003 0 0 ) ( 0.8826 0 0 )
( 0 1/b 0 ) = ( 0 -0.801 0 ), ( 0 3.003 0 ), ( 0 0.8826 0 )
( 0 0 1/b) ( 0 0 -0.801) ( 0 0 3.003 ) ( 0 0 0.8826 ).