JEE Advanced 2021 · Paper 1 · Mathematics Q14

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MathematicsMultiple correct+4 / -2Question 52 of 57

For any $3\times3$ matrix $M$ , let $|M|$ denote the determinant of $M$ . Let $I$ be the $3\times3$ identity matrix. Let $E$ and $F$ be two $3\times3$ matrices such that $(I - EF)$ is invertible. If $G = (I - EF)^{-1}$ , then which of the following statements is (are) TRUE?

|FE| = |I - FE|\,|FGE|

(I - FE)(I + FGE) = I

EFG = GEF

(I - FE)(I - FGE) = I

Answer and worked solution

Official answer: A, B, C

Two forms of G. From $G(I-EF)=I$ and $(I-EF)G=I$ :
$G=I+GEF=I+EFG.$
So $EFG=GEF$ , and (C) is true.

Option (B).
$(I-FE)(I+FGE)=I-FE+F(G-EFG)E=I-FE+FE=I.$
True.

Option (A). By (B), $(I-FE)^{-1}=I+FGE$ , so
$FGE=(I-FE)^{-1}-I=(I-FE)^{-1}FE.$
Taking determinants, $|FGE|=\dfrac{|FE|}{|I-FE|}$ . True.

Option (D). $(I-FE)(I-FGE)=I-FE-F(G-EFG)E=I-2FE\ne I$ in general. False.

(A), (B), (C)

Topics:Adjoint and inverse Matrix algebra Determinants and properties

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JEE Advanced 2021 · Paper 1, Mathematics question 14