QJEEAdv21Ma4033: JEE Advanced 2021 Mathematics - Class 12 JEEAdv21Ma4033 Solution: For any 3 × 3 matrix M, let

Question

Verified Solution • 2025–26 Session moderate

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?

Accepted Answer

∵

I

-

EF = G^{$-$ 1}

\Rightarrow

G

-

GEF = I ..... (i)

and G

-

EFG = I ..... (ii)

Clearly, GEF = EFG

\to

option (c) is correct.

Also, (I

-

FE) (I + FGE)

= I

-

FE + FGE

-

FEFGE

= I

-

FE + FGE

-

F(G

-

I) E

= I

-

FE + FGE

-

FGE + FE

= I

\to

option (b) is correct but option (d) is incorrect.

∵

(I

-

FE) (I

-

FGE) = I

-

FE

-

FGE + F(G

-

I) E

= I

-

2FE

Now, (I

-

FE) (

-

FGE) =

-

FE

\Rightarrow

| I

-

FE | | FGE | = | FE |

\to

option (a) is correct.

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