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

Example 13  - Chapter 4 Class 12 Determinants - Part 2

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Question 8 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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.