JEE Advance - Mathematics (2021 - Paper 1 Online - No. 14)
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?
| FE | = | I $$-$$ FE| | FGE |
(I $$-$$ FE)(I + FGE) = I
EFG = GEF
(I $$-$$ FE)(I $$-$$ FGE) = I
Explanation
$$\because$$ 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.
$$\because$$ (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.
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.
$$\because$$ (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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