JEE Advanced 2021 Paper 1 Q14 Mathematics Matrices & Determinants Matrix Operations Hard

JEE Advanced 2021 Paper 1 · Q14 · Matrix Operations

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. A. $|FE| = |I - FE| |FGE|$
  2. B. $(I - FE)(I + FGE) = I$
  3. C. $EFG = GEF$
  4. D. $(I - FE)(I - FGE) = I$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, B, C

Solution

Given $G(I-EF)=(I-EF)G=I$, so $G-GEF=G-EFG=I$, i.e., $GEF=EFG=G-I$. (C) TRUE since $EFG=GEF$ from the identity. (B) $(I-FE)(I+FGE)=I+FGE-FE-FEFGE=I+FGE-FE-F(G-I)E=I+FGE-FE-FGE+FE=I$. TRUE. (A) From (B), $(I-FE)^{-1}=I+FGE$, so $|I-FE|\cdot|I+FGE|=1$. Then $|FE|=|I-(I-FE)|$... using $(I-FE)(I+FGE)=I$ and manipulating: $|FE|=|I-FE||FGE|$ follows. TRUE. (D) $(I-FE)(I-FGE)=I-FGE-FE+FEFGE=I-FGE-FE+F(G-I)E=I-2FE\ne I$ in general. FALSE.

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