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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

If Δ=|[ a1 b1...

Question

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 $∣ ∣ ∣ ∣ \begin{matrix} A_{1} & B_{1} & C_{1} A_{2} & B_{2} & C_{2} A_{3} & B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ ∣$ is

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Δ²

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Δ³

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Solution

The correct option is B Δ²
We know that $Δ Δ^{^{'}} = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} A_{1} & B_{1} & C_{1} A_{2} & B_{2} & C_{2} A_{3} & B_{3} & C_{3} \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} \sum a_{1} A_{1} & 0 & 0 0 & \sum a_{2} A_{2} & 0 0 & 0 & \sum a_{3} A_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} Δ & 0 & 0 0 & Δ & 0 0 & 0 & Δ \end{matrix} ∣ ∣ ∣ ∣ = Δ^{3} \Rightarrow Δ^{^{'}} = Δ^{2} .$

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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} . . . .

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} ∣ ∣ ∣ ∣,

then

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

Q. If in 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} ∣ ∣ ∣ ∣, A_{1}, B_{1}, C_{1}

etc. be the co-factors of

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

etc., then which of the following relations is incorrect

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 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} ∣ ∣ ∣ ∣

is-

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 ?

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If Δ = ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣ and A1, B1, C1 denote the co-factors of a1, b1, c1 respectively, then the value of the determinant ∣∣ ∣∣A1B1C1A2B2C2A3B3C3∣∣ ∣∣ is

The correct option is B Δ2We know that ΔΔ′ = ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣ ∣∣ ∣∣A1B1C1A2B2C2A3B3C3∣∣ ∣∣ =∣∣ ∣ ∣∣∑ a1A1000∑ a2A2000∑ a3A3∣∣ ∣ ∣∣ = ∣∣ ∣∣Δ000Δ000Δ∣∣ ∣∣ = Δ3⇒ Δ′ = Δ2.