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Question 15

Using properties of determinants, prove that:

|a (b - c) (c + b) (a + c) b (c - a) (a - b) (b + a) c| = (a + b + c) (a 2 + b 2 + c 2 )


Transcript

Question 15 Using properties of determinants, prove that: |■8(𝑎&𝑏−𝑐&𝑐+𝑏@𝑎+𝑐&𝑏&𝑐−𝑎@𝑎−𝑏&𝑏+𝑎&𝑐)| = (𝑎+𝑏+𝑐) (𝑎^2+𝑏^2+𝑐^2) Solving LHS |■8(𝑎&𝑏−𝑐&𝑐+𝑏@𝑎+𝑐&𝑏&𝑐−𝑎@𝑎−𝑏&𝑏+𝑎&𝑐)| Since we cannot solve by adding or subtracting rows We multiply a with C1, b with C2 and c with C3 = 𝑎𝑏𝑐/𝑎𝑏𝑐 |■8(𝑎&𝑏−𝑐&𝑐+𝑏@𝑎+𝑐&𝑏&𝑐−𝑎@𝑎−𝑏&𝑏+𝑎&𝑐)| = 1/𝑎𝑏𝑐 |■8(𝑎(𝑎)&𝑏(𝑏−𝑐)&𝑐(𝑐+𝑏)@𝑎(𝑎+𝑐)&𝑏(𝑏)&𝑐(𝑐−𝑎)@𝑎(𝑎−𝑏)&𝑏(𝑏+𝑎)&𝑐(𝑐))| = 1/𝑎𝑏𝑐 |■8(𝑎^2&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@𝑎^2+𝑎𝑐&𝑏^2&𝑐^2−𝑎𝑐@𝑎^2−𝑎𝑏&𝑏^2+𝑎𝑏&𝑐^2 )| C1 → C1 + C2 + C3 = 1/𝑎𝑏𝑐 |■8(𝑎^2+𝑏^2−𝑏𝑐+𝑐^2+𝑏𝑐&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@𝑎^2+𝑎𝑐+𝑏^2+𝑐^2−𝑎𝑐&𝑏^2&𝑐^2−𝑎𝑐@𝑎^2−𝑎𝑏+𝑏^2+𝑎𝑏+𝑐^2&𝑏^2+𝑎𝑏&𝑐^2 )| = 1/𝑎𝑏𝑐 |■8(𝑎^2+𝑏^2+𝑐^2&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@𝑎^2+𝑏^2+𝑐^2&𝑏^2&𝑐^2−𝑎𝑐@𝑎^2+𝑏^2+𝑐^2&𝑏^2+𝑎𝑏&𝑐^2 )| Taking 𝑎^2+𝑏^2+𝑐^2 common from C1 = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@1&𝑏^2&𝑐^2−𝑎𝑐@1&𝑏^2+𝑎𝑏&𝑐^2 )| R2 → R2 – R1 = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@1−1&𝑏^2−(𝑏^2−𝑏𝑐)&𝑐^2−𝑎𝑐−(𝑐^2+𝑏𝑐)@1&𝑏^2+𝑎𝑏&𝑐^2 )| = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@0&𝑏𝑐&−𝑎𝑐−𝑏𝑐@1&𝑏^2+𝑎𝑏&𝑐^2 )| R3 → R3 – R1 = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@0&𝑏𝑐&−𝑎𝑐−𝑏𝑐@1−1&𝑏^2+𝑎𝑏−(𝑏^2−𝑏𝑐)&𝑐^2−(𝑐^2+𝑏𝑐))| = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏^2−𝑏𝑐&𝑐^2+𝑏𝑐@0&𝑏𝑐&−𝑎𝑐−𝑏𝑐@0&𝑎𝑏+𝑏𝑐&−𝑏𝑐)| = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐 |■8(1&𝑏(𝑏−𝑐)&𝑐(𝑐+𝑏)@0&𝑏𝑐&−𝑐(𝑎+𝑏)@0&𝑏(𝑎+𝑐)&−𝑏𝑐)| Taking b common from C2, and c common from C3 = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎𝑏𝑐×𝑏𝑐|■8(1&(𝑏−𝑐)&(𝑐+𝑏)@0&𝑐&−(𝑎+𝑏)@0&(𝑎+𝑐)&−𝑏)| = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 |■8(1&(𝑏−𝑐)&(𝑐+𝑏)@0&𝑐&−(𝑎+𝑏)@0&(𝑎+𝑐)&−𝑏)| = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 |■8(1&(𝑏−𝑐)&(𝑐+𝑏)@0&𝑐&−(𝑎+𝑏)@0&(𝑎+𝑐)&−𝑏)| Finding determinant along C1 = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × [1 (c × (-b) – (–(a + b))(a + c) + 0 + 0] = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × [–bc + (a + b)(a + c)] = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × [–bc + a(a + c) + b(a + c)] = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × [–bc + a2 + ac + ab + bc] = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × [a2 + ac + ab] = ((𝑎^2 + 𝑏^2+ 〖 𝑐〗^2))/𝑎 × a[a + c + b] = (𝑎+𝑏+𝑐) (𝑎^2+𝑏^2+𝑐^2)

  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

About the Author

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo