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Example 30  - If a, b, c are positive, unequal, show determinant - Examples

Example 30  - Chapter 4 Class 12 Determinants - Part 2
Example 30  - Chapter 4 Class 12 Determinants - Part 3 Example 30  - Chapter 4 Class 12 Determinants - Part 4

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Example 30 If a, b, c are positive and unequal, show that value of the determinant Δ = a﷮b﷮c﷮b﷮c﷮a﷮c﷮a﷮b﷯﷯ is negative Δ = a﷮b﷮c﷮b﷮c﷮a﷮c﷮a﷮b﷯﷯ Applying C1→ C1 + C2 + C3 = 𝐚+𝐛+𝐜﷮b﷮c﷮𝐚+𝐛+𝐜﷮c﷮a﷮𝐚+𝐛+𝐜﷮a﷮b﷯﷯ Taking common a + b + c from C1 = (𝐚+𝐛+𝐜) 1﷮b﷮c﷮1﷮c﷮a﷮1﷮a﷮b﷯﷯ Applying R2 → R2 – R1 = (a+b+c) 1﷮b﷮c﷮𝟏−𝟏﷮c−b﷮a−c﷮1﷮a﷮b﷯﷯ = (a+b+c) 1﷮b﷮c﷮𝟎﷮c−b﷮a−c﷮1﷮a﷮b﷯﷯ Applying R3 → R2 – R1 = (a+b+c) 1﷮b﷮c﷮0﷮c−b﷮a−c﷮𝟏−𝟏﷮a−b﷮b−c﷯﷯ = (a+b+c) 1﷮b﷮c﷮0﷮c−b﷮a−c﷮𝟎﷮a−b﷮b−c﷯﷯ Expanding determinant along C1 = (a + b + c ) 1 c−b﷮a−c﷮b−c﷮b−c﷯﷯−0 b﷮c﷮𝑎−𝑏﷮b−c﷯﷯+0 b﷮c﷮c−b﷮a−c﷯﷯﷯ = (a + b + c ) 1 𝑐−𝑎﷯ 𝑏−𝑐﷯− 𝑎−𝑏﷯ 𝑎−𝑐﷯−0+0﷯ = (a + b + c ) 𝑏−𝑐﷯ − 𝑏−𝑐﷯﷯−(𝑎−𝑏)(𝑎−𝑐)﷯ = (a + b + c ) − 𝑏2+𝑐2−2𝑏𝑐﷯−(𝑎2−𝑎𝑐−𝑏𝑎+𝑏𝑐)﷯ = (a + b + c ) −a2−b2 −𝑐2+𝑎𝑏+𝑏𝑐+𝑐𝑎﷯ = – (a + b + c ) a2+b2+𝑐2−𝑎𝑏−𝑏𝑐−𝑐𝑎﷯ Multiplying & Dividing by 2 = – 1 × 2﷮2﷯ (a + b + c ) a2+b2+𝑐2−𝑎𝑏−𝑏𝑐−𝑐𝑎﷯ = −1﷮2﷯ (a + b + c ) 2a2+2b2+2𝑐2−2𝑎𝑏−2𝑏𝑐−2𝑐𝑎﷯ = −1﷮2﷯ (a + b + c ) 𝑎2+𝑎2+𝑏2+𝑏2+𝑐2+𝑐2−2𝑎𝑏−2𝑏𝑐−2𝑐𝑎﷯ = −1﷮2﷯ (a + b + c ) 𝒂𝟐+𝒄𝟐−𝟐𝒄𝒂+𝑎2+𝑏2−2𝑎𝑏+𝑏2+𝑐2−2𝑏𝑐﷯ = −1﷮2﷯ (a + b + c ) 𝒂−𝒄﷯𝟐+ 𝑎−𝑏﷯2+ 𝑐−𝑎﷯2﷯ Now 𝑎−𝑐﷯2+ 𝑎−𝑏﷯2+ 𝑐−𝑎﷯2 > 0 & a + b + c > 0 ∴ ∆ = −1﷮2﷯(a + b + c ) 𝑎−𝑐﷯2+ 𝑎−𝑏﷯2+ 𝑐−𝑎﷯2 < 0 Hence ∆ is negative Hence Shown

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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.