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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 1 : 0
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Detailed Solution

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Concept:

Multiplying the elements of any row with the cofactors of another row is 0.

Calculation:

If C11, C12, C13 be the cofactors of a11, a12, a13 respectively,

⇒ C11 = a22a33 - a23a32, C12 = a23a31 - a21a33 ,C13 = a21a32 - a22a31,

Putting the values in a21C11 + a22C12 + a23C13

⇒ a21C11 + a22C12 + a23C13 = a21(a22a33 - a23a32) + a22(a23a31 - a21a33) + a23(a21a32 - a22a31)

⇒ a21C11 + a22C12 + a23C13 = a21a22a33 - a21a23a32 + a22a23a31 - a21a22a33 + a23a21a32 - a22a23a31

⇒ a21C11 + a22C12 + a23C13 = 0

∴ The correct answer is option (1).

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