For any $3 \times 3$ matrix $M$, let $|M|$ 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?

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If G = (I - EF)⁻¹, then which of the following statements is (are) TRUE? $\text{Given: } G = (I - EF)^{-1},\quad I - EF \text{ invertible},\quad |M| = \det M$ $\text{To find: which of (1)--(4) are TRUE}$ $1.\ \text{from } G = (I-EF)^{-1}:\ EFG,\ GEF$ $2.\ \text{determinant identity for (1)}$ $3.\ \text{expand the products for (2), (4)}$ $(I - EF)G = I \quad \text{and} \quad G(I - EF) = I$ $(I-EF)G = I \Rightarrow G - EFG = I \Rightarrow EFG = G - I$ $G(I-EF) = I \Rightarrow G - GEF = I \Rightarrow GEF = G - I$ $\Rightarrow EFG = GEF \quad\Rightarrow\ \text{(3) TRUE}$ $|I - EF| = |I - FE|$ $|G| = \dfrac{1}{|I - EF|} = \dfrac{1}{|I - FE|}$ $|FGE| = |F|\,|G|\,|E| = |FE|\,|G| = \dfrac{|FE|}{|I - FE|}$ $\Rightarrow\ |FE| = |I - FE|\,|FGE| \quad\Rightarrow\ \text{(1) TRUE}$ $FE\,FGE = F(EFG)E = F(G-I)E = FGE - FE$ $(I - FE)(I + FGE) = I + FGE - FE - FE\,FGE$ $= I + FGE - FE - (FGE - FE) = I \quad\Rightarrow\ \text{(2) TRUE}$ $(I - FE)(I - FGE) = I - FGE - FE + FE\,FGE$ $= I - FGE - FE + (FGE - FE) = I - 2FE \neq I$ $\Rightarrow\ \text{(4) FALSE}$ $\boxed{\text{True: } (1),\ (2),\ (3)}$