# Question 6 [JEE adv, 2021]

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?

1. $$|FE| = |I-FE||FGE|$$
2. $$(I-FE)(I + FGE) = I$$
3. $$EFG = GEF$$
4. $$(I - FE)(I-FGE) = I$$

## Concepts

Use $$AA^{-1} = A^{-1}A = I$$

## Solution

#### Step 1

$\begin{eqnarray} (I - EF)G &=& G(I-EF)\\ \implies EFG &=& GEF \end{eqnarray}$ Option 3 is correct.

#### Step 2

We have that \begin{aligned} G(I - EF) &= I\implies GEF = G-I\\ (I - EF)G & = I\implies EFG = G -I \end{aligned} Now, \begin{aligned} (I-FE)(I + FGE) &= I - FE + FGE - FEFGE\\ &= I - FE + FGE - F(G - I)E\\ &= I - FE + FGE - FGE + FE\\ &= I \end{aligned} Option 2 is correct.

#### Step 3

\begin{aligned} (I-FE)(I - FGE) &= I - FE - FGE + FEFGE\\ &= I - FE - FGE + F(G - I)E\\ &= I - FE + FGE + FGE - FE\\ &= I - 2FE \end{aligned} Option 4 is incorrect.

#### Step 4

\begin{aligned} (I-FE)FGE &= FGE - FEFGE\\ &= FGE - F(G - I)E\\ &= FGE - FGE + FE\\ &= FE \end{aligned} Option 1 is correct.