# Misc. 16 - Chapter 4 Class 12 Determinants

Last updated at Jan. 23, 2020 by Teachoo

Last updated at Jan. 23, 2020 by Teachoo

Transcript

Misc 16 Solve the system of the following equations 2/x + 3/y + 10/z = 4 4/x + 6/y + 5/z = 1 6/x + 9/y + 20/z = 2 The system of equations are 2/x + 3/y + 10/z = 4 4/x + 6/y + 5/z = 1 6/x + 9/y + 20/z = 2 Now let ๐/๐ = u , ๐/๐ = v , & ๐/๐ = w The system of equations become 2u + 3v + 10w = 4 4u โ 6v + 5w = 1 6u + 9v โ 20w = 2 Writing equation as AX = B [โ 8(2&3&10@4&โ6&5@6&9&โ20)] [โ 8(๐ข@๐ฃ@๐ค)] = [โ 8(4@1@2)] Hence A = [โ 8(2&3&10@4&โ6&5@6&9&โ20)] , X = [โ 8(๐ข@๐ฃ@๐ค)] & B = [โ 8(4@1@2)] Calculating |A| |A| = |โ 8(2&3&10@4&โ6&5@6&9&โ20)| = 2 |โ 8(โ6&5@9&โ20)| โ 3 |โ 8(4&5@6&โ20)| + 10 |โ 8(4&โ6@6&9)| = 2 (120 โ 45) โ3 (โ80 โ 30) + 10 ( 36 + 36) = 2 (75) โ3 (โ110) + 10 (72) = 150 + 330 + 720 = 1200 โด |A|โ 0 So, the system of equation is consistent & has a unique solution Now, AX = B X = A-1 B Calculating A-1 Now, A-1 = 1/(|A|) adj (A) adj (A) = [โ 8(A11&A12&A13@A21&A22&A23@A31&A32&A33)]^โฒ = [โ 8(A11&A21&A31@A12&A22&A32@A13&A23&A33)] A = [โ 8(2&3&10@4&โ6&5@6&9&โ20)] M11 = |โ 8(โ6&5@9&โ20)| = 120 โ 45 = 75 M12 = |โ 8(4&5@6&โ20)| = (โ80 โ 30) = โ110 M13 = |โ 8(4&โ6@6&9)| = 36 โ36 = 72 M21 = |โ 8(3&10@9&โ20)| = โ60 โ 90 = โ150 M22 = |โ 8(2&10@6&โ20)| = โ40 โ 60 = โ100 M23 = |โ 8(2&3@6&9)| = 18 โ 18 = 0 M31 = |โ 8(3&10@โ6&5)| = 15 + 60 = 75 M32 = |โ 8(2&10@4&5)| = 10 โ 40 = โ30 M33 = |โ 8(2&3@4&โ6)| = โ12 โ 12 = โ24 Now, A11 = ใ"(โ1)" ใ^(1+1) M11 = (โ1)2 . 75 = 75 A12 = ใ"(โ1)" ใ^"1+2" M12 = ใ"(โ1)" ใ^3 . (โ110) = 110 A13 = ใ(โ1)ใ^(1+3) M13 = ใ(โ1)ใ^4 . (72) = 72 A21 = ใ(โ1)ใ^(2+1) M21 = ใ(โ1)ใ^3 . (โ150) = 150 A22 = ใ(โ1)ใ^(2+2) M22 = (โ1)4 . (โ100) = โ100 A23 = ใ(โ1)ใ^(2+3). M23 = ใ(โ1)ใ^5. 0 = 0 A31 = ใ(โ1)ใ^(3+1). M31 = ใ(โ1)ใ^4 . 75 = 75 A32 = ใ(โ1)ใ^(3+2) . M32 = ใ(โ1)ใ^5. (โ30) = 30 A33 = ใ(โ1)ใ^(3+3) . M33 = (โ1)6 . โ24 = โ24 Thus, adj A = [โ 8(75&150&75@110&โ110&30@72&0&โ24)] Now, A-1 = 1/(|A|) adj A A-1 = 1/1200 [โ 8(75&150&75@110&โ110&30@72&0&โ24)] Also, X = Aโ1 B Putting Values [โ 8(๐ข@๐ฃ@๐ค)]= 1/1200 [โ 8(75&150&75@110&โ110&30@72&0&โ24)] [โ 8(4@1@2)] [โ 8(๐ข@๐ฃ@๐ค)]= 1/1200 [โ 8(75(4)+150(1)+75(4)@110(4)+(โ110)(1)+30(1)@72(4)+0(1)+(โ24)2)] [โ 8(๐ข@๐ฃ@๐ค)] = 1/1200 [โ 8(300+150+150@440โ100+60@288+0โ48)] = 1/1200 [โ 8(600@400@140)] [โ 8(๐ข@๐ฃ@๐ค)] = [โ 8(1/2@1/3@1/5)] Hence u = 1/2 , v = 1/3 , & w = 1/5 Thus, x = 2, y = 3 & z = 5 Putting u = ๐/๐ 1/2 = 1/๐ฅ x = 2 Putting v = ๐/๐ 1/3 = 1/๐ฆ y = 3 Putting w = ๐/๐ 1/5 = 1/๐ง z = 5

Miscellaneous

Misc 1

Misc. 2 Important Deleted for CBSE Board 2021 Exams only

Misc 3

Misc 4

Misc 5

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11 Important Deleted for CBSE Board 2021 Exams only

Misc 12 Important Deleted for CBSE Board 2021 Exams only

Misc. 13 Deleted for CBSE Board 2021 Exams only

Misc 14 Deleted for CBSE Board 2021 Exams only

Misc. 15 Important Deleted for CBSE Board 2021 Exams only

Misc. 16 Important You are here

Misc 17 Important Deleted for CBSE Board 2021 Exams only

Misc 18

Misc 19 Important

Matrices and Determinants - Formula Sheet and Summary

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.