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If  then what is A(adj A) equal to?

(where) I is the identity matrix.

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  1. Null matrix
  2. -I
  3. I
  4. 2I

Answer (Detailed Solution Below)

Option 4 : 2I
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Detailed Solution

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)Concept:

Expansion of determinant

  is given by 

Δ = a11 a22 a33 - a11 a23 a32 - a12 a21 a33 + a12 a23 a31 + a13 a21 a32 - a13 a31 a22

A(adj A) = ∣ A ∣ I, where I is the identity matrix.

Calculation:

Given:

   and I is the identity matrix.

⇒ 

Expanding about C3, we get,

⇒ ∣ A ∣ =  1{(2 sinθ ×  sinθ) + 2 cos2θ}

⇒ ∣ A ∣ =  1(2 sin2θ + 2 cos2 θ) 

⇒ ∣ A ∣ =  2(sin2θ + cos2θ)

⇒ ∣ A ∣ =  2          [∵ sin2θ + cos2θ = 1]

⇒ A(adj A) = ∣ A ∣ I

Putting the value of ∣ A ∣ =  2,

⇒ A(adj A) =  2I, where I is the identity matrix.

∴ A(adj A) =  2I

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