1. Tardigrade
  2. Question
  3. Mathematics
  4. 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?

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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?

JEE AdvancedJEE Advanced 2021

Solution:

$G ( I - EF )=( I - EF ) G = I$
$\Rightarrow G - GEF = G - EFG = I$ ... (1)
(A) $| FE |=| I - FE || FGE |=| FGE - FE FGE |$
$=| FGE - F ( G - I ) E |=| FGE - FGE + FE |=| FE |$
(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) $( I - FE )( I - FGE )= I - FGE - FE + FEFGE$
$= I - FGE - FE + F ( G - I ) E = I - FGE - FE + FGE - FE$ $= I -2 FE$