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Let M = 0 1 a 1 2 3 3 b 1 and a d j   M = - 1 1 - 1 8 - 6 2 - 5 3 - 1 where a and b are real numbers. Which of the following options is/are correct?

Options

  1. Aa + b = 3
  2. Bd e t a d j M 2 = 81
  3. Ca d j M - 1 + a d j M - 1 = - M
  4. DIf M = α β γ = 1 2 3 , then α - β + γ = 3

Correct answer

A. a + b = 3

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 ) = (

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