Question

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?

• A. $|FE|=|I-FE||FGE|$
• B. $(I-FE)(I+FGE)=I$
• C. $EFG=GEF$
• D. $(I-FE)(I-FGE)=I$

Answer 1

Answer: (A), (B), (C)

$G(I-EF)=(I-EF)G=I$
$\Rightarrow G-GEF=G-EFG=I$ ... (1)
(A) $|FE|=|I-FE||FGE|=|FGE-FEFGE|$
$=|FGE-F(G-I)E|=|FGE-FGE+FE|=|FE|$
(B) $(I-FE)(I+FGE)=I+FGE-FE-FEFGH$
$=I+FGE-FE-F(G-I)E=I+FGE-FE-FGE+FE=I$
(C) From (I) it is true
(D) $(I-FE)(I-FGE)=I-FGE-FE+FEFGE$
$=I-FGE-FE+F(G-I)E=I-FGE-FE+FGE-FE$ $=I-2FE$