If A1- B1- C1 - are the cofactors of the elements a1- b1- c1- of the matrix - 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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Cofactor

If A1, B1, C1...

Question

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 $∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ =$

a1 D

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a1a3D

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(a1 +a2)D

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None of these

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Solution

The correct option is A
a1 D

Here as per the statement of the question B2, C2, B3 and C3 are the co factors of b2,c2 b3 and c3 respectively. So let’s calculate them first.

$B 2 = (- 1)^{2 + 2} ∣ ∣ ∣ \begin{matrix} a 1 & c 1 a 3 & c 3 \end{matrix} ∣ ∣ ∣ = (a 1 c 3 - c 1 a 3)$

$C 2 = (- 1)^{2 + 3} ∣ ∣ ∣ \begin{matrix} a 1 & b 1 a 3 & b 3 \end{matrix} ∣ ∣ ∣ = - (a 1 b 3 - a 3 b 1)$

$B 3 = (- 1)^{3 + 2} ∣ ∣ ∣ \begin{matrix} a 1 & c 1 a 2 & c 2 \end{matrix} ∣ ∣ ∣ = - (a 1 c 2 - a 2 c 1)$

$C 3 = (- 1)^{3 + 3} ∣ ∣ ∣ \begin{matrix} a 1 & b 1 a 2 & b 2 \end{matrix} ∣ ∣ ∣ = (a 1 b 2 - a 2 b 1)$

$∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} a 1 c 3 - a 3 c 1 & - (a 1 b 3 - a 3 b 1) - (a 1 c 2 - a 2 c 1) & a 1 b 2 - a 2 b 1 \end{matrix} ∣ ∣ ∣$

Solving this we will get

$a 1^{2} (b 2 c 3 - b 3 c 2) + a 1 b 1 (a 3 c 2 - c 3 a 2) + a 1 c 1 (- a 3 b 2 + a 2 b 3) + c 1 b 1 (a 3 a 2 - a 2 a 3)$

$= a 1 (a 1 (b 2 c 3 - b 3 c 2) + b 1 (a 3 c 2 - c 3 a 2) + c 1 (- a 3 b 2 + a 2 b 3) - ---------------------------------------------------------- - + 0)$

Now if you look carefully, the underlined part is the value of the determinant D expanded along the first row

So, $∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ = a 1 D$ which is the option (a)

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Similar questions

Q. If

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

are respectively the co-factors of the elements

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

Q. If

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

and

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

denote the co-factors 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} ∣ ∣ ∣ ∣

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 $∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ =$

Q. 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 ?

Q. 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

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

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If A1, B1, C1 … are the cofactors of the elements a1, b1, c1... of the matrix ⎡⎢⎣a1b1c1a2b2c2a3b3c3⎤⎥⎦ then ∣∣∣B2C2B3C3∣∣∣=

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 $∣ ∣ ∣ \begin{matrix} B 2 & C 2 B 3 & C 3 \end{matrix} ∣ ∣ ∣ =$