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.