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 ?

JEE Advanced 2021 Mathematics question

For any matrix , let denote the determinant of . Let be the identity matrix. Let and be two matrices such that is invertible. If , then which of the following statements is (are) TRUE ?

(A)Correct
(B)Correct
(C)Correct
(D)

Answer: (A),(B),(C)

Solution

From we get , and from we get . (B) (I-FE)(I+FGE) = I-FE+FGE-FEFGE = I-FE+F(G-EFG)E = I-FE+F(I)E = I – TRUE. (D) then fails, because FE)^{-1}$ = I+FGE≠

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Question ID VK-033891