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If A1,B1,C1...

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 C $a_{1} Δ$
$B_{2} = a_{1} c_{3} - a_{3} c_{1}, C_{2} = - (a_{1} b_{3} - a_{3} b_{1}); B_{3} = - (a_{1} c_{2} - a_{2} c_{1}), C_{3} = a_{1} b_{2} - a_{2} b_{1}$

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

$= ∣ ∣ ∣ \begin{matrix} a_{1} c_{3} - a_{3} c_{1} & - (a_{1} b_{3} - a_{3} b_{1}) - (a_{1} c_{2} - a_{2} c_{1}) & a_{1} b_{2} - a_{2} b_{1} \end{matrix} ∣ ∣ ∣$

$= a_{1}^{2} ∣ ∣ ∣ \begin{matrix} c_{3} & - b_{3} - c_{2} & b_{2} \end{matrix} ∣ ∣ ∣ + a_{1} b_{1} ∣ ∣ ∣ \begin{matrix} {- c}_{3} & a_{3} c_{2} & {- a}_{2} \end{matrix} ∣ ∣ ∣ + a_{1} c_{1} ∣ ∣ ∣ \begin{matrix} {- a}_{3} & {- b}_{3} a_{2} & b_{2} \end{matrix} ∣ ∣ ∣ + b_{1} c_{1} ∣ ∣ ∣ \begin{matrix} {- a}_{3} & a_{3} a_{2} & {- a}_{2} \end{matrix} ∣ ∣ ∣$

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

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Similar questions

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} ∣ ∣ ∣

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}

denote the cofactors of

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

respectively, then the value of the determinant

∣ ∣ ∣ ∣ \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} . . . .

are respectively the co-factors 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} ∣ ∣ ∣ ∣,

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

If A1, B1, C1 … are the cofactors of the elements a1, b1, c1... of the matrix

$⎡ ⎢ ⎣ \begin{matrix} a 1 & b 1 & c 1 a 2 & b 2 & c 2 a 3 & b 3 & c 3 \end{matrix} ⎤ ⎥ ⎦$ then $∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ =$

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}

denote the co-factors of

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

respectively, then teh value os the determinant

∣ ∣ ∣ ∣ \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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