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Let Δ be the determinant of a matrix A, where A = and C11, C12, C13 be the cofactors of a11, a12, a13 respectively.

What is the value of  ?

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NDA 02/2022 Mathematics Official Paper (Held On 04 Sep 2022)
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  1. 0
  2. 1
  3. Δ
  4. −Δ

Answer (Detailed Solution Below)

Option 4 : −Δ
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Detailed Solution

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Concept:

  • By interchanging two rows or columns determinant changes to its negative value.
  • The determinant of the transpose of a matrix is same as the determinant of A.

Calculation:

If  Δ = |A| = ,

Taking transpose of A,

⇒ |AT| =   = Δ ,

Take C1 ↔ C3,

⇒   = - Δ 

Take C1 ↔ C2,

⇒   = -(- Δ) = Δ

Take R2 ↔ R3,

⇒   = - Δ

∴ The correct option is (4).

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