Question
If A1, B1, C1.... are respectively the co-factors of the elements a1, b1, c1.... of the determinant Δ = ∣∣
∣∣a1b1c1a2b2c2a3b3c3∣∣
∣∣, then ∣∣∣B2C2B3C3∣∣∣ =
∣∣a1b1c1a2b2c2a3b3c3∣∣
∣∣, then ∣∣∣B2C2B3C3∣∣∣ =
Answer: Option A
:
A
B2=∣∣∣a1c1a3c3∣∣∣=a1c3−c1a3C2=−∣∣∣a1b1a3b3∣∣∣=−(a1b3−a3b1)B3=−∣∣∣a1c1a2c2∣∣∣=−(a1c2−a2c1)C3=∣∣∣a1b1a2b2∣∣∣=(a1b2−a2b1)∣∣∣B2C2B3C3∣∣∣=∣∣∣a1c3−a3c1−(a1b3−a3b1)−(a1c2−a2c1)a1b2−a2b1∣∣∣=∣∣∣a1c3−a1b3−a1c2a1b2∣∣∣+∣∣∣a1c3a3b1−a1c2−a2b1∣∣∣+∣∣∣−a3c1−a1b3a2c1a1b2∣∣∣+∣∣∣−a3c1a3b1a2c1−a2b1∣∣∣=a12(b2c3−b3c2)+a1b1(−c3a2+a3c2)+a1c1(−a3b2+a2b3)+c1b1(a3a2−a2a3)=a1Δ.
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:
A
B2=∣∣∣a1c1a3c3∣∣∣=a1c3−c1a3C2=−∣∣∣a1b1a3b3∣∣∣=−(a1b3−a3b1)B3=−∣∣∣a1c1a2c2∣∣∣=−(a1c2−a2c1)C3=∣∣∣a1b1a2b2∣∣∣=(a1b2−a2b1)∣∣∣B2C2B3C3∣∣∣=∣∣∣a1c3−a3c1−(a1b3−a3b1)−(a1c2−a2c1)a1b2−a2b1∣∣∣=∣∣∣a1c3−a1b3−a1c2a1b2∣∣∣+∣∣∣a1c3a3b1−a1c2−a2b1∣∣∣+∣∣∣−a3c1−a1b3a2c1a1b2∣∣∣+∣∣∣−a3c1a3b1a2c1−a2b1∣∣∣=a12(b2c3−b3c2)+a1b1(−c3a2+a3c2)+a1c1(−a3b2+a2b3)+c1b1(a3a2−a2a3)=a1Δ.
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