For any 3 × 3 matrix M , let | M | denote the determinant of M ....

(A) $G = ( I - EF{)}^{- 1}$

$⇒ FGE = F ( I - EF{)}^{- 1}E$

$= {\left({E}^{- 1} ( I - EF ) {F}^{- 1}\right)}^{- 1}$

$⇒ FGE = {\left({E}^{- 1}{F}^{- 1} - I\right)}^{- 1}$

$⇒ | FGE | \cdot \left|{E}^{- 1}{F}^{- 1} - I\right| = 1$

$⇒ | FGE | \cdot \left|{E}^{- 1}{F}^{- 1} - I\right| | FE | = | FE |$

$⇒ | FGE | | I - FE | = | FE |$

(B) & (D) $( I - FE ) \cdot ( I + FGE )$

$= I + FGE - FE - FE \cdot FGE = 0$

$= I + FGE - FE - F ( G - I ) E$

$= I + FGE - FE - FGE + FE = I$

(C) $( I - EF ) G = I = G ( I - EFG )$

$⇒ G - EFG = I = G - GEF$

$⇒ EFG = GEF$