Whole row/column one

Chapter 4 Class 12 Determinants
Concept wise

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Question 8 Evaluate โ = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฏ๏ทฏ โ = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฏ๏ทฏ We need to obtain 0 in 2nd row as well as 3rd row Applying R2 โ R2 โ R1 โ = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ๐ โ๐๏ทฎ๐โ๐๏ทฎ๐๐โ๐๐๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฏ๏ทฏ = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ๐๏ทฎ๐โ๐๏ทฎ๐(๐โ๐)๏ทฎ1๏ทฎ๐๏ทฎ๐๐๏ทฏ๏ทฏ Applying R3 โ R3 โ R1 = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ0๏ทฎ๐โ๐๏ทฎ๐(๐โ๐)๏ทฎ๐โ๐๏ทฎ๐โ๐๏ทฎ๐๐โ๐๐๏ทฏ๏ทฏ = 1๏ทฎ๐๏ทฎ๐๐๏ทฎ0๏ทฎ(๐โ๐)๏ทฎ๐(๐โ๐)๏ทฎ๐๏ทฎ(๐โ๐)๏ทฎ๐(๐โ๐)๏ทฏ๏ทฏ Expanding it along C1 = 1 ๐โ๐๏ทฏ๏ทฎ๐ ๐โ๐๏ทฏ๏ทฎ ๐โ๐๏ทฏ๏ทฎ๐ ๐โ๐๏ทฏ๏ทฏ๏ทฏโ0 ๐๏ทฎ๐๐๏ทฎ ๐โ๐๏ทฏ๏ทฎ๐ ๐โ๐๏ทฏ๏ทฏ๏ทฏ +0 ๐๏ทฎ๐๐๏ทฎ ๐โ๐๏ทฏ๏ทฎ๐ ๐โ๐๏ทฏ๏ทฏ๏ทฏ = 1 bโ๐๏ทฏ๏ทฎc ๐โ๐๏ทฏ๏ทฎ cโa๏ทฏ๏ทฎ๐ ๐โ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 โ = (a โ b) (b โ c) (c โ a)