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Cofactor

If A 1 , B ...

Question

If $A_{1}, B_{1}, C_{1}, . .$ are, respectively, the cofactors of the elements $a_{1}, b_{1}, c_{1}, . .$ of the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ , $Δ \neq 0$ , then the value of $∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣$ is equal to

A

{a_{1}}^{2} Δ

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B

a_{1} Δ

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C

a_{1} Δ^{2}

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D

{a_{1}}^{2} Δ^{2}

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Solution

The correct option is B $a_{1} Δ$
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow Δ = a_{1} (b_{2} c_{3} - b_{3} c_{2}) + b_{1} (a_{3} c_{2} - a_{2} c_{3}) + c_{1} (a_{2} b_{3} - a_{3} b_{2})$
We have to find $∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣$
Here, $B_{2}$ is cofactor of element $b_{2}$
So, $B_{2} = ∣ ∣ ∣ \begin{matrix} a_{1} & c_{1} a_{3} & c_{3} \end{matrix} ∣ ∣ ∣$
$\Rightarrow B_{2} = a_{1} c_{3} - a_{3} c_{1}$
$C_{2}$ is cofactor of element $c_{2}$
So, $C_{2} = - ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} a_{3} & b_{3} \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{2} = a_{3} b_{1} - a_{1} b_{3}$
$B_{3}$ is cofactor of element $b_{3}$
So, $B_{3} = - ∣ ∣ ∣ \begin{matrix} a_{1} & c_{1} a_{2} & c_{2} \end{matrix} ∣ ∣ ∣$
$\Rightarrow B_{3} = a_{2} c_{1} - a_{1} c_{2}$
$C_{3}$ is cofactor of element $c_{3}$
So, $C_{3} = ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} a_{2} & b_{2} \end{matrix} ∣ ∣ ∣$
$\Rightarrow C_{3} = a_{1} b_{2} - a_{2} b_{1}$
Now, $∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣$
$= ∣ ∣ ∣ \begin{matrix} a_{1} c_{3} - a_{3} c_{1} & a_{3} b_{1} - a_{1} b_{3} a_{2} c_{1} - a_{1} c_{2} & a_{1} b_{2} - a_{2} b_{1} \end{matrix} ∣ ∣ ∣$
$= {a_{1}}^{2} b_{2} c_{3} - a_{1} a_{2} b_{1} c_{3} - a_{1} a_{3} b_{2} c_{1} - {a_{1}}^{2} b_{3} c_{2} + a_{1} a_{2} b_{3} c_{1} + a_{1} a_{3} b_{1} c_{2}$
$= a_{1} [a_{1} (b_{2} c_{3} - b_{3} c_{2}) + b_{1} (a_{3} c_{2} - a_{2} c_{3}) + c_{1} (a_{2} b_{3} - a_{3} b_{2})]$
$= a_{1} Δ$

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