If A1- B1- C1- are respectively the co-factor of the elements a1- b1- c1- - a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3- then B 2 C 2 B 3 C 3

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If A1, B1, ...

Question

If $A_{1}, B_{1}, C_{1} . .$ are respectively the co-factor of the elements $a_{1}, b_{1}, c_{1}$ .
$△ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ , then $∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣$

a_{1} △

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a_{1} a_{3} △

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(a_{1} + b_{1}) △

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N o n e o f t h e s e

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Solution

The correct option is A $a_{1} △$
$\begin{matrix} B_{2} = a_{1} c_{3} - a_{3} c_{1} c_{2} = - (a_{1} b_{3} - a_{3} b_{1}) B_{3} = - (a_{1} c_{2} - a_{2} c_{1}) c_{3} = a_{1} b_{2} - b_{1} a_{2} ∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} a_{1} c_{3} - a_{3} c_{1} & - a_{1} b_{3} + a_{3} b_{1} - a_{1} c_{2} + a_{2} c_{1} & a_{1} b_{2} - b_{1} a_{2} \end{matrix} ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} a_{1} c_{3} & - a_{1} b_{3} - a_{1} c_{2} & a_{1} b_{2} \end{matrix} ∣ ∣ ∣ + ∣ ∣ ∣ \begin{matrix} a_{1} c_{3} & a_{3} b_{1} - a_{1} c_{2} & - a_{2} b_{1} \end{matrix} ∣ ∣ ∣ + ∣ ∣ ∣ \begin{matrix} - a_{3} c_{1} & - a_{1} b_{3} a_{2} c_{1} & a_{1} b_{2} \end{matrix} ∣ ∣ ∣ 1 ∣ ∣ ∣ \begin{matrix} - a_{3} c_{1} & a_{3} b_{1} a_{2} c_{1} & - a_{2} b_{1} \end{matrix} ∣ ∣ ∣ = a_{1}^{2} ∣ ∣ ∣ \begin{matrix} c_{3} & - b_{3} - c_{2} & b_{2} \end{matrix} ∣ ∣ ∣ + a_{1} b_{1} ∣ ∣ ∣ \begin{matrix} c_{3} & a_{3} - c_{2} & - a_{2} \end{matrix} ∣ ∣ ∣ + a_{1} c ∣ ∣ ∣ \begin{matrix} - a_{3} & - b_{3} a_{2} & b_{2} \end{matrix} ∣ ∣ ∣ + b_{1} c_{1} ∣ ∣ ∣ \begin{matrix} - a_{3} & a_{3} a_{2} & - a_{2} \end{matrix} ∣ ∣ ∣ = a_{1} {a_{1} (b_{2} c_{3} - b_{3} c_{2}) - b_{1} (a_{2} c_{3} - a_{3} c_{2}) + c_{1} (a_{2} b_{3} - a_{3} b_{2})} = a_{1} ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = a_{1} Δ \end{matrix}$

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Q. If

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

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

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

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is-

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If A1,B1,C1.. are respectively the co-factor of the elements a1,b1,c1.△=∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣, then ∣∣∣B2C2B3C3∣∣∣