Since (I−EF) is invertible therefore, |I−EF|≠0;G=(I−EF)−1⇒G−1=I−EF We have G⋅G−1=I=G−1G ⇒G(I−EF)=I=(I−EF)G ⇒G−GEF=I=G−EFG......(i) ⇒GEF=EFG( Option (c) is correct ) (I−FE)(I+FGE)=I+FGE−FE−FEFGE =I+FGE−FE−F(G−I)E[ from (i) ] =I+FGE−FE−FGE+FE =I[ Option (b) is correct ] (I−FE)(I+FGE)=I..........(ii) (So, option 'd' is incorrect) Now FE(I+FGE) =FE+FEFGE =FE+F(G−I)E[ form (i) ] =FE+FGE−FE=FGE ⇒|FE||I+FGE|=|FGE| ⇒|FE|×
1
|I−FE|
=|FGE| (from(ii)) ⇒|FE|=|I−FE||FGE| (Option (a) is correct)