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Evaluation of a Determinant

If in the det...

Question

If in the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣, A_{1}, B_{1}, C_{1}$ etc. be the co-factors of $a_{1}, b_{1}, c_{1}$ 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 $a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2} = Δ$
This is how a determinant is evaluated. If we are evaluating a determinant along a particular row or a column, each element of that row or column is multiplied with its cofactor. And then they are added.

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Similar questions

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}

a_{1}, b_{1}, c_{1}

∣ ∣ ∣ ∣ \begin{matrix} A_{1} & B_{1} & C_{1} A_{2} & B_{2} & C_{2} A_{3} & B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ ∣

⎡ ⎢ ⎣ \begin{matrix} b_{1} + c_{1} & c_{1} + a_{1} & a_{1} + b_{1} b_{2} + c_{2} & c_{2} + a_{2} & a_{2} + b_{2} b_{3} + c_{3} & c_{3} + a_{3} & a_{3} + b_{3} \end{matrix} ⎤ ⎥ ⎦ =

A_{1}, B_{1}, C_{1} . . . .

a_{1}, b_{1}, c_{1} . . . .

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣,

∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ =

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}

a_{1}, b_{1}, c_{1}

A_{1}, B_{1}, C_{1} . . . .

a_{1}, b_{1}, c_{1} . . . .

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣,

∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ =

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