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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&[email protected]&b2&[email protected]&c2&ab)| = |■8(1&a2&[email protected]&b2&[email protected]&c2&c3)| Solving L.H.S |■8(a&a2&[email protected]&b2&[email protected]&c2&ab)| Multiplying and dividing by abc = abc/abc |■8(a&a2&[email protected]&b2&[email protected]&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&[email protected]&b3&[email protected]&c3&1)| = |■8(a2&a3&[email protected]&b3&[email protected]&c3&1)| Interchange C1 ↔ C3 = (–1) |■8(1&a3&[email protected]&b3&[email protected]&c3&c2)| Interchange C2 ↔ C3 = (–1) (–1) |■8(1&a2&[email protected]&b2&[email protected]&c2&c3)| = |■8(1&a2&[email protected]&b2&[email protected]&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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.