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?
A
| FE | = | I $$-$$ FE| | FGE |
B
(I $$-$$ FE)(I + FGE) = I
C
EFG = GEF
D
(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.
2
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
For any 3 $$\times$$ 3 matrix M, let | M | denote the determinant of M. Let
From Eqs. (iv) and (v) option (d) is also correct.
3
JEE Advanced 2020 Paper 1 Offline
MCQ (More than One Correct Answer)
Let M be a 3 $$ \times $$ 3 invertible matrix with real entries and let I denote the 3 $$ \times $$ 3 identity matrix. If M$$-$$1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?
A
M = I
B
det M = 1
C
M2 = I
D
(adj M)2 = I
Explanation
It is given that matrix M be a 3 $$ \times $$ 3 invertible matrix, such that
M$$-$$1 = adj(adj M) $$ \Rightarrow $$ M$$-$$1 = |M| M
($$ \because $$ for a matrix A of order 'n' adj(adjA) = |A|n$$-$$2 A}
$$ \Rightarrow $$ M$$-$$1 M = |M|M2 $$ \Rightarrow $$ M2|M| = I .....(i)