For any  $3\times 3$ matrix M, let  $\left|M\right|$ denote the determinant of M. Let I be the  $3\times 3$ identity matrix. Let E and F be two  $3\times 3$ matrices such that (I – EF) is invertible. If $G={(I-EF)}^{-1}$, then which of the following statements is (are) TRUE ?

 $|I-EF|\ne 0;G={(I-EF)}^{-1}\Rightarrow G^{-1}=I-EF$
Now,  $G.G^{-1}=I=G^{-1}G$
 
  $$\begin{gathered} \Rightarrow G(I-EF)=I=(I-EF)G \\ \Rightarrow G-GEF=I=G-EFG \\ \Rightarrow GEF=EFG \end{gathered}$$
 (I – FE) (I + FGE)   = I + FGE – FE – FEFGE
 = I + FGE – FE – F (G – I) E
 = I + FGE – FE – FGE + FE
 = I
(I – FE) (I + FGE) = I …… (I)
Now,
FE (I + FGE)
= FE + FEFGE
= FE + F(G – I)E
= FE + FGE – FE
= FGE
 $\Rightarrow \left|FE\right||I+FGE|=\left|FGE\right|$
  $\Rightarrow \left|FE\right|\times \frac{1}{|I-FE|}=\left|FGE\right|$(from (1))
 $\Rightarrow \left|FE\right|=|I-FE|\left|FGE\right|$