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)

$| FE |=| I - FE || FGE |$

Answer 2

$I – EF = G^{–1}$

$G – GEF = I …(1)$

And $G – EFG = I …(2)$

Clearly $GEF = EFG$ (option C is correct)

Also $(I – FE)(I + FGE) = I – FE + FGE – FE + FGE$

$= I – FE + FGE – F(G – I)E$

$= I – FE + FGE – FGE + FE$

$= I$ (option B is correct and D is incorrect)

Now, $(I – FE)(I – FGE) = I – FE – FGE + F(G – I)E$

$= I – 2FE$

$(I – FE)(- FGE) = – FE$

$|I – FE||FGE| = |FE|$