If $△ = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ⎤ ⎥ ⎦$ and $A_{2}, B_{2}, C_{2}$ are respectively cofactors of $a_{2}, b_{2}, c_{2}$ then $a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2}$ is equal to ?

- △

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Solution

The correct option is B $0$
Co-factor of $a_{2} = (- 1)^{i + j} ∣ ∣ ∣ \begin{matrix} b_{1} & c_{1} b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ = - [b_{1} c_{3} - c_{1} b_{3}] = A_{2}$
i=2 j=1

of $b_{2} = (- 1)^{2 + 2} ∣ ∣ ∣ \begin{matrix} a_{1} & c_{1} a_{3} & c_{3} \end{matrix} ∣ ∣ ∣ = [a_{1} c_{3} - c_{1} a_{3}] = B_{2}$

of $c_{2} = (- 1)^{2 + 3} ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} a_{3} & b_{3} \end{matrix} ∣ ∣ ∣ = - [a_{1} b_{3} - b_{1} a_{3}] = C_{2}$

$a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2}$

$- a_{1} b_{1} c_{3} + a_{1} c_{1} b_{3} + b_{1} a_{1} c_{3} - b_{1} c_{1} a_{3} - c_{1} a_{1} b_{3} + c_{1} b_{1} a_{3}$

$= 0$

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