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Question

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

a1b1c1a2b2c2a3b3c3 then B2C2B3C3=


A

a1 D

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B

a1a3D

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C

(a1 +a2)D

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D

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.

B2=(1)2+2a1c1a3c3=(a1c3c1a3)

C2=(1)2+3a1b1a3b3=(a1b3a3b1)

B3=(1)3+2a1c1a2c2=(a1c2a2c1)

C3=(1)3+3a1b1a2b2=(a1b2a2b1)

B2C2B3C3=a1c3a3c1(a1b3a3b1)(a1c2a2c1)a1b2a2b1

Solving this we will get

a12(b2c3b3c2)+a1b1(a3c2c3a2)+a1c1(a3b2+a2b3)+c1b1(a3a2a2a3)

=a1(a1(b2c3b3c2)+b1(a3c2c3a2)+c1(a3b2+a2b3)––––––––––––––––––––––––––––––––––––––––––––––––––––––––––+0)

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

So, B2C2B3C3=a1 D which is the option (a)


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