Misc. 2 - Class 12

Transcript

Misc. 2 Without expanding the determinant, prove that a﷮a2﷮bc﷮b﷮b2﷮ca﷮c﷮c2﷮ab﷯﷯ = 1﷮a2﷮a3﷮1﷮b2﷮b3﷮1﷮c2﷮c3﷯﷯ Taking L.H.S a﷮a2﷮bc﷮b﷮b2﷮ca﷮c﷮c2﷮ab﷯﷯ Multiplying and dividing by abc = abc﷮abc﷯ a﷮a2﷮bc﷮b﷮b2﷮ca﷮c﷮c2﷮ab﷯﷯ Multiplying a to R1, b to R2 & c to R3 = 1﷮abc﷯ a(𝑎)﷮𝑎(a2)﷮a(bc)﷮b(𝑏)﷮b(b2)﷮b (ca)﷮c 𝑐﷯﷮𝑐(c2)﷮c (ab)﷯﷯ Multiplying a to R1, b to R2 & c to R3 = 1﷮abc﷯ a(𝑎)﷮𝑎(a2)﷮a(bc)﷮b(𝑏)﷮b(b2)﷮b (ca)﷮c 𝑐﷯﷮𝑐(c2)﷮c (ab)﷯﷯ = 1﷮abc﷯ a2﷮a3﷮𝑎𝑏𝑐﷮b2﷮b3﷮𝑎𝑏𝑐﷮c2﷮c3﷮𝑎𝑏𝑐﷯﷯ Taking abc common from C3 = 𝑎𝑏𝑐﷮𝑎𝑏𝑐﷯ a2﷮a3﷮1﷮b2﷮b3﷮1﷮c2﷮c3﷮1﷯﷯ = a2﷮a3﷮1﷮b2﷮b3﷮1﷮c2﷮c3﷮1﷯﷯ Interchange C1 ↔ C3 = ( – 1) 1﷮a3﷮a2﷮1﷮b3﷮b2﷮1﷮c3﷮c2﷯﷯ Interchange C2 ↔ C3 = ( – 1) ( – 1) 1﷮a2﷮a3﷮1﷮b2﷮b3﷮1﷮c2﷮c3﷯﷯ = 1﷮a2﷮a3﷮1﷮b2﷮b3﷮1﷮c2﷮c3﷯﷯ = R.H.S. Hence Proved.