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Misc 2 - Without expanding the determinant, prove that - Miscellaneous

Misc. 2 - Chapter 4 Class 12 Determinants - Part 2
Misc. 2 - Chapter 4 Class 12 Determinants - Part 3

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Misc 2 Without expanding the determinant, prove that |■8(a&a2&bc@b&b2&ca@c&c2&ab)| = |■8(1&a2&a3@1&b2&b3@1&c2&c3)| Solving L.H.S |■8(a&a2&bc@b&b2&ca@c&c2&ab)| Multiplying and dividing by abc = abc/abc |■8(a&a2&bc@b&b2&ca@c&c2&ab)| Multiplying a to R1, b to R2 & c to R3 = 1/abc |■8(a(𝑎)&𝑎(a2)&a(bc)@b(𝑏)&b(b2)&b (ca)@c(𝑐)&𝑐(c2)&c (ab))| Multiplying a to R1, b to R2 & c to R3 = 1/abc |■8(a(𝑎)&𝑎(a2)&a(bc)@b(𝑏)&b(b2)&b (ca)@c(𝑐)&𝑐(c2)&c (ab))| = 1/abc |■8(a2&a3&𝑎𝑏𝑐@b2&b3&𝑎𝑏𝑐@c2&c3&𝑎𝑏𝑐)| Taking abc common from C3 = 𝑎𝑏𝑐/𝑎𝑏𝑐 |■8(a2&a3&1@b2&b3&1@c2&c3&1)| = |■8(a2&a3&1@b2&b3&1@c2&c3&1)| Interchange C1 ↔ C3 = (–1) |■8(1&a3&a2@1&b3&b2@1&c3&c2)| Interchange C2 ↔ C3 = (–1) (–1) |■8(1&a2&a3@1&b2&b3@1&c2&c3)| = |■8(1&a2&a3@1&b2&b3@1&c2&c3)| = R.H.S. Hence Proved We know that If any two row or column of a determinant are interchanged, then sign of determinant changes.

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.