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Let Δ be the determinant of a matrix A, where A = \(\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]\) and C11, C12, C13 be the cofactors of a11, a12, a13 respectively.

What is the value of a11C11 + a12C12 + a13C13 ?

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  1. 0
  2. 1
  3. Δ
  4. −Δ

Answer (Detailed Solution Below)

Option 3 : Δ
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Detailed Solution

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Concept:

Multiplying the elements of any row with their respective cofactors gives the determinant of the matrix.

Calculation:

Given matrix A = \(\left(\begin{array}{lll}{a}_{11}&{a}_{12}&{a}_{13}\\{a}_{21}&{a}_{22}&{a}_{23}\\{a}_{31}&{a}_{32}&{a}_{33}\end{array}\right)\) 

Expanding along row 1,

|A| = Δ = a11(a22a33 - a23a32) + a12(a23a31 - a21a33) + a

13(a21a32 - a22a31)

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

where, C11, C12, C13 are the cofactors of a11, a12, a13 respectively.

Putting the values,

⇒ a11C11 + a12C12 + a13C13 = Δ

∴ The correct answer is option (3).

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