Example 13 - Evaluate determinant |1 a bc 1 b ca 1 c ab| - Whole row/column one

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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise
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Example 13 Evaluate โˆ† = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘๐‘Ž๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘Ž๐‘๏ทฏ๏ทฏ โˆ† = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘๐‘Ž๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘Ž๐‘๏ทฏ๏ทฏ We need to obtain 0 in 2nd row as well as 3rd row Applying R2 โ†’ R2 โ€“ R1 โˆ† = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ๐Ÿ โ€“๐Ÿ๏ทฎ๐‘โˆ’๐‘Ž๏ทฎ๐‘๐‘Žโˆ’๐‘๐‘๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘Ž๐‘๏ทฏ๏ทฏ = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ๐ŸŽ๏ทฎ๐‘โˆ’๐‘Ž๏ทฎ๐‘(๐‘Žโˆ’๐‘)๏ทฎ1๏ทฎ๐‘๏ทฎ๐‘Ž๐‘๏ทฏ๏ทฏ Applying R3 โ†’ R3 โ€“ R1 = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ0๏ทฎ๐‘โˆ’๐‘Ž๏ทฎ๐‘(๐‘Žโˆ’๐‘)๏ทฎ๐Ÿโˆ’๐Ÿ๏ทฎ๐‘โˆ’๐‘Ž๏ทฎ๐‘Ž๐‘โˆ’๐‘๐‘๏ทฏ๏ทฏ = 1๏ทฎ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ0๏ทฎ(๐‘โˆ’๐‘Ž)๏ทฎ๐‘(๐‘Žโˆ’๐‘)๏ทฎ๐ŸŽ๏ทฎ(๐‘โˆ’๐‘Ž)๏ทฎ๐‘(๐‘Žโˆ’๐‘)๏ทฏ๏ทฏ Expanding it along C1 = 1 ๐‘โˆ’๐‘Ž๏ทฏ๏ทฎ๐‘ ๐‘Žโˆ’๐‘๏ทฏ๏ทฎ ๐‘โˆ’๐‘Ž๏ทฏ๏ทฎ๐‘ ๐‘Žโˆ’๐‘๏ทฏ๏ทฏ๏ทฏโ€“0 ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ ๐‘โˆ’๐‘Ž๏ทฏ๏ทฎ๐‘ ๐‘Žโˆ’๐‘๏ทฏ๏ทฏ๏ทฏ +0 ๐‘Ž๏ทฎ๐‘๐‘๏ทฎ ๐‘โˆ’๐‘Ž๏ทฏ๏ทฎ๐‘ ๐‘Žโˆ’๐‘๏ทฏ๏ทฏ๏ทฏ = 1 bโˆ’๐‘Ž๏ทฏ๏ทฎc ๐‘Žโˆ’๐‘๏ทฏ๏ทฎ cโˆ’a๏ทฏ๏ทฎ๐‘ ๐‘Žโˆ’๐‘๏ทฏ๏ทฏ๏ทฏ โ€“ 0 + 0 = 1 (b โ€“ a) b(a โ€“ c) โ€“ (c โ€“ a) (c) (a โ€“ b)๏ทฏ = โ€“(a โ€“ b) b ( โ€“ (c โ€“ a)) โ€“ (c โ€“ a) c (a โ€“ b) = (a โ€“ b) b (c โ€“ a) โ€“ (c โ€“ a) c (a โ€“ b) = (a โ€“ b)(c โ€“ a) (b โ€“ c)) = (a โ€“ b) (b โ€“ c) (c โ€“ a) Thus โˆ† = (a โ€“ b) (b โ€“ c) (c โ€“ a)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.