Ex 4.2, 8 - Show using property (i) |1 a a2 1 b b2 1 c ca|

Ex 4.2, 8 - Chapter 4 Class 12 Determinants - Part 2
Ex 4.2, 8 - Chapter 4 Class 12 Determinants - Part 3

 

  1. Chapter 4 Class 12 Determinants (Term 1)
  2. Serial order wise

Transcript

Ex 4.2, 8 By using properties of determinants, show that: (i) |โ– 8(1&๐‘Ž&๐‘Ž2@1&๐‘&๐‘2@1&๐‘&๐‘2)| = (a - b) (b - c)(c โ€“ a) Solving L.H.S |โ– 8(1&๐‘Ž&๐‘Ž2@1&๐‘&๐‘2@1&๐‘&๐‘2)| Applying R1 โ†’ R1 โˆ’ R2 = |โ– 8(๐Ÿโˆ’๐Ÿ&๐‘Žโˆ’๐‘&๐‘Ž^2โˆ’๐‘^2@1&๐‘&๐‘2@1&๐‘&๐‘2 ) | = |โ– 8(๐ŸŽ&(๐‘Žโˆ’๐‘)&(๐‘Žโˆ’๐‘)(๐‘Ž+๐‘)@1&๐‘&๐‘2@1&๐‘&๐‘2 ) | = |โ– 8(0(๐šโˆ’๐›)&(๐šโˆ’๐›)&(๐šโˆ’๐’ƒ)(a+b)@1&b&b2@1&c&c2 ) | Taking Common (a โ€“ b) from R1 = (๐šโˆ’๐’ƒ) |โ– 8(0&1&a+b@1&b&b2@1&c&c2 ) | Applying R2 โ†’ R2 โˆ’ R3 = (aโˆ’b) |โ– 8(0&1&a+b@๐Ÿโˆ’๐Ÿ&bโˆ’c&b2โˆ’c2@1&c&c2 ) | = (a โ€“ b) |โ– 8(0&1&a+๐‘@๐ŸŽ&bโˆ’c&(bโˆ’c)(b+c)@1&c&c2 ) | Taking common (b โ€“ c) from R2 = (a โ€“ b) (b โ€“ c) |โ– 8(0&1&a+b@0&1&b+c@1&c&c2 ) | Expanding Determinant along C1 = (a โ€“ b) (b โ€“ c) ( 0|โ– 8(1&๐‘+๐‘@๐‘&๐‘2)|โˆ’0|โ– 8(1&๐‘Ž+๐‘@๐‘&๐‘2)|+1|โ– 8(1&๐‘Ž+๐‘@1&๐‘+๐‘)|) = (a โ€“ b) (b โ€“ c) ( 0โˆ’0+1|โ– 8(1&๐‘Ž+๐‘@1&๐‘+๐‘)|) = (a โ€“ b) (b โ€“ c) (1(b + c) โ€“ 1(a + b) ) = (a โ€“ b) (b โ€“ c) (b + c โ€“ a โ€“ b) = (a โ€“ b) (b โ€“ c)(c โ€“ a) = R.H.S Hence Proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.