Matrices — JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 Question with Solution

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 - E F)^{- 1}

, then which of the following statements is (are) TRUE?

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

Solution

Given

G = (I - E F)^{- 1}

, we have

G^{- 1} = I - E F

, so

G (I - E F) = I = (I - E F) G

. This gives

G - GE F = I = G - E F G

, hence

GE F = E F G

(C is correct). From

G (I - E F) = I

, we get

G - GE F = I

, so

GE F = G - I

. Now

(I - F E) (I + F GE) = I + F GE - F E - F E F GE = I + F GE - F E - F (G - I) E = I + F GE - F E - F GE + F E = I

(B is correct, D is incorrect). For (A):

F E (I + F GE) = F E + F E F GE = F E + F (G - I) E = F E + F GE - F E = F GE

. Taking determinants:

∣ F E ∣∣ I + F GE ∣ = ∣ F GE ∣

. From (B),

∣ I + F GE ∣ = \frac{1}{∣ I - F E ∣}

, so

∣ F E ∣ \cdot \frac{1}{∣ I - F E ∣} = ∣ F GE ∣

, giving

∣ F E ∣ = ∣ I - F E ∣∣ F GE ∣

(A is correct).

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · One or more correct options