JEE Advanced Mathematics Question (2019) — Solution

Step-by-step solution

Given, M = 0 1 a 1 2 3 3 b 1 ...(i) a d j M = - 1 1 - 1 8 - 6 2 - 5 3 - 1 ...(ii) Option ( A ) : Now from matrix M a d j   M 11 = 2 - 3 b As given 2 - 3 b = - 1 ⇒ b = 1 Similarly,  a = 2 a + b = 3 Using the values of a and b in matrix M ⇒ | M | = - 2 | a d j M 2 |   =   | M 2 | 2 = | M | 4 = 16 Now, M - 1 = a d j   M M a d j   M = M M - 1 a d j   ( M ) =   - 2 M - 1 a d j M - 1 = - 1 2 M - 1 - 1 K A - 1 = 1 K A - 1 ( a d j ( M ) ) - 1 = - M 2 and a d j ( M - 1 ) = ( M - 1 ) - 1 | M - 1 | a d j   A = A A 1 a d j M - 1 = M M A - 1 = 1 A a d j M - 1 = - M 2 ...(iii) From (iii) and (iv) ( a d j   M - 1 + a d j   M - 1 = - M Option ( D ) : M X = B       w h e n   x = α β γ   a n d   B = 1 2 3 X = M - 1   ( B ) X = a d j M M B α β γ = - 1 2 - 1 1 - 1 8 - 6 2 - 5 3 - 1 1 2 3 = - 1 2 - 2 2 - 2 α β γ = 1 - 1 1 ∵   α = 1     β = - 1     γ = 1 α - β + γ = 3