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+2∣∣∣a1c1a3c3∣∣∣=(a1c3−c1a3) C2=(−1)2+3∣∣∣a1b1a3b3∣∣∣=−(a1b3−a3b1) B3=(−1)3+2∣∣∣a1c1a2c2∣∣∣=−(a1c2−a2c1) C3=(−1)3+3∣∣∣a1b1a2b2∣∣∣=(a1b2−a2b1) ∣∣∣B2C2B3C3∣∣∣=∣∣∣a1c3−a3c1−(a1b3−a3b1)−(a1c2−a2c1)a1b2−a2b1∣∣∣ Solving this we will get a12(b2c3−b3c2)+a1b1(a3c2−c3a2)+a1c1(−a3b2+a2b3)+c1b1(a3a2−a2a3) =a1(a1(b2c3−b3c2)+b1(a3c2−c3a2)+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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