Properties of Invertible Matrix Products and Identities

To determine which of the given statements are true, we use the given matrix equation $G = (I - EF)^{-1}$.

Since $G$ is the inverse of $(I - EF)$, we have: $(I - EF)G = G(I - EF) = I$

Expanding both products:

1. $(I - EF)G = I \implies G - EFG = I \implies EFG = G - I$
2. $G(I - EF) = I \implies G - GEF = I \implies GEF = G - I$

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Analysis of Option (C):

From the relations above, we have: $EFG = G - I \quad \text{and} \quad GEF = G - I$ Thus, $EFG = GEF$ Hence, Option (C) is TRUE.

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Analysis of Option (B):

Consider the expression $(I - FE)(I + FGE)$: $(I - FE)(I + FGE) = I + FGE - FE - FEFGE$

Substitute $EFG = G - I$ into the product $FEFGE = F(EFG)E$: $FEFGE = F(G - I)E = FGE - FE$

Substituting this back into the expansion gives: $(I - FE)(I + FGE) = I + FGE - FE - (FGE - FE) = I$ Hence, Option (B) is TRUE.

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Analysis of Option (A):

Consider the matrix product $(I - FE)(FGE)$: $(I - FE)(FGE) = FGE - FEFGE = FGE - F(EFG)E$

Using $EFG = G - I$: $(I - FE)(FGE) = FGE - F(G - I)E = FGE - (FGE - FE) = FE$

Taking the determinant on both sides: $|FE| = |(I - FE)(FGE)|$

Using the property $|AB| = |A||B|$ for any square matrices $A$ and $B$: $|FE| = |I - FE||FGE|$ Hence, Option (A) is TRUE.

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Analysis of Option (D):

Consider the product $(I - FE)(I - FGE)$: $(I - FE)(I - FGE) = I - FGE - FE + FEFGE$

Substitute $FEFGE = FGE - FE$: $(I - FE)(I - FGE) = I - FGE - FE + (FGE - FE) = I - 2FE \neq I$ Hence, Option (D) is FALSE.

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Conclusion:

The correct statements are (A), (B), and (C).