If in the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣, A_{i}, B_{i}, C_{i}$ etc. be the co-factors of $a_{i}, b_{i}, c_{i}$ etc., then which of the following relations is incorrect?

a_{1} A_{1} + b_{1} B_{1} + c_{1} C_{1} = Δ

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a_{2} A_{2} + b_{2} B_{2} + c_{2} C_{2} = Δ

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a_{3} A_{3} + b_{3} B_{3} + c_{3} C_{3} = Δ

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a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2} = Δ

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Solution

The correct option is D $a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2} = Δ$
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣, A_{1}, B_{1}, C_{1}$ etc. be the co-factors of $a_{1}, b_{1}, c_{1}$ etc.
Sum of product of any row with its corresponding cofactors is Determinant.
$\Rightarrow a_{1} A_{1} + b_{1} B_{1} + c_{1} C_{1} = a_{2} A_{2} + b_{2} B_{2} + c_{2} C_{2} = a_{3} A_{3} + b_{3} B_{3} + c_{3} C_{3} = Δ$
Sum of product of any row with another row's cofactors is $0$ .
$\Rightarrow a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2} = 0$
Hence, option D is incorrect.

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