If - a1 b1 c1- a2 b2 c2- a3 b3 c3 -x then - a1 b1 c1- a2 b2 c2- a3 b3 c3 - a1 a2 a3- b1 b2 b3- c1 c2 c3 -

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Byju's Answer

Standard XII

Mathematics

Transpose of a Matrix

If |[ a1 b1 c...

Question

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

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

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Solution

The correct option is B 2x
Here the second matrix is the transpose of the first matrix. When we take determinant of transpose of a matrix it comes out to be the same as the original determinant. This is one of the properties of determinants. So in short both the first determinant and the second determinant are equal to each other.

Hence, if $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = x$ ,
then
$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ + $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 2 x$

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Similar questions

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

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} ∣ ∣ ∣ ∣

Q. Let

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

and

Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} α_{1} & β_{1} & γ_{1} α_{2} & β_{2} & γ_{2} α_{3} & β_{3} & γ_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

, then

Δ_{1} \times Δ_{2}

can be expressed as the sum of how many determinants?

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 5, t h e n t h e v a l u e o f ∣ ∣ ∣ ∣ \begin{matrix} b_{2} c_{3} - b_{3} c_{2} & a_{3} c_{2} - a_{2} c_{3} & a_{2} b_{3} - a_{3} b_{2} b_{3} c_{1} - b_{1} c_{3} & a_{1} c_{3} - a_{3} c_{1} & a_{3} b_{1} - a_{1} b_{3} b_{1} c_{2} - b_{2} c_{1} & a_{2} c_{1} - a_{1} c_{2} & a_{1} b_{2} - a_{2} b_{1} \end{matrix} ∣ ∣ ∣ ∣ i s$

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} ∣ ∣ ∣ =

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If ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣= x then ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣ + ∣∣ ∣∣a1a2a3b1b2b3c1c2c3∣∣ ∣∣ =