Prove the following : ∣∣
∣∣bcbc′+b′cb′c′caca′+c′ac′a′abab′+d′bd′b′∣∣
∣∣=(bc′−b′c)(ca′−c′a)(ab′−db)
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Solution
Hint : Multiply R1,R2,R3 by a,b,c respectively and divide the det. by abc. Now operate R2−R1 and R3−R1. This will give $\Delta =\cfrac { 1 }{ abc } \begin{vmatrix} abc & abc'+ab'c & ab'c' \\ 0 & c\left( db-b'a \right) & c'\left( bd-ab' \right) \\ 0 & b\left( dc-c'a \right) & b'\left( c