If the co factors of the elements a1,b1,c1 .... are A1,B1,C1,...are respectively then the value of the determinant Δ=a1 b1 c1a2 b2 c2a3 b3 c3, Δ≠0, then the value of B2 C2B3 C3 is equal to
a12Δ
a1Δ
a1Δ2
a12Δ2
B2=a1c3−a3c1,C2=−a1b3−a3b1B3=−a1c2−a2c1,C3=a1b2−a2b1
∴ B2C2B3C3=a1c3−a3c1−a1b3+a3b1−a1c2+a2c1a1b2−a2b1=a1c3−a1b3−a1c2a1b2+a1c3a3b1−a1c2−a2b1+−a3c1−a1b3a2c1a1b2+−a3c1a3b1a2c1−a2b1
=a12c3−b3−c2b2+a1b1c3a3−c2−a2+a1c1−a3−b3a2b2+b1c1−a3a3a2−a2
=a1a1b2c3−b3c2−b1a2c3−a3c2+c1a2b3−a3b2
=a1a1 b1 c1a2 b2 c2a3 b3 c3=a1Δ