# Misc. 16

Last updated at March 20, 2018 by Teachoo

Last updated at March 20, 2018 by Teachoo

Transcript

Misc. 16 Solve the system of the following equations 2x + 3y + 10z = 4 4x + 6y + 5z = 1 6x + 9y + 20z = 2 The system of equations are 2x + 3y + 10z = 4 4x + 6y + 5z = 1 6x + 9y + 20z = 2 Now let 𝟏𝒙 = u , 𝟏𝒚 = v , & 𝟏𝒛 = w The system of equations become 2u + 3v + 10w = 4 4u – 6v + 5w = 1 6u + 9v – 20w = 2 Step 1 Write equation as AX = B 23104−6569−20 𝑢𝑣𝑤 = 412 Hence A = 23104−6569−20 , X = 𝑢𝑣𝑤 & B = 412 Step 2 Calculate |A| |A| = 23104−6569−20 = 2 −659−20 – 3 456−20 + 10 4−669 = 2 ( 120 – 45) – 3 ( – 80 – 30) + 10 ( 36 + 36) = 2 (75) – 3 (– 110) + 10 (72) = 150 + 330 + 720 = 1200 ∴ |A|≠ 0 So, the system of equation is consistent & has a unique solution Now, AX = B X = A-1 B Step 3 Calculating X = A-1 B Now, A-1 = 1|A| adj (A) adj (A) = A11A12A13A21A22A23A31A32A33′ = A11A21A31A12A22A32A13A23A33 A = 23104−6569−20 M11 = −659−20 = 120 – 45 = 75 M12 = 456−20 = ( – 80 – 30) = – 110 M13 = 4−669 = 36 – 36 = 72 M21 = 3109−20 = − 60 – 90 = – 150 M22 = 2106−20 = – 40 – 60 = – 100 M23 = 2369 = 18 – 18 = 0 M31 = 310−65 = 15 + 60 = 75 M32 = 21045 = 10 – 40 = – 30 M33 = 234−6 = – 12 – 12 = – 24 A11 = ( –1)1+1 M11 = ( –1)2 . 75 = 75 A12 = ( –1)1+2 M12 = ( –1)3 . ( – 110) = 110 A13 = ( −1)1+3 M13 = ( −1)4 . (72) = 72 A21 = ( −1)2+1 M21 = ( −1)3 . ( – 150) = 150 A22 = ( −1)2+2 M22 = ( –1)4 . ( – 100) = – 100 A23 = (−1)2+3. M23 = (−1)5. 0 = 0 A31 = (−1)3+1. M31 = (−1)4 . 75 = 75 A32 = (−1)3+2 . M32 = (−1)5. ( – 30) = 30 A33 = (−1)3+3 . M33 = ( –1)6 . – 24 = – 24 Thus adj A = 7515075110−11030720−24 Now, A-1 = 1|A| adj A A-1 = 11200 7515075110−11030720−24 X = A-1 B Putting Values 𝑢𝑣𝑤= 11200 7515075110−11030720−24 412 𝑢𝑣𝑤= 11200 75(4)+150(1)+75(4)110(4)+(−110)(1)+30(1)72(4)+0(1)+ −242 𝑢𝑣𝑤 = 11200 300+150+150440−100+60288+0−48 = 11200 600400140 ∴ 𝑢𝑣𝑤 = 12 13 15 Hence u = 12 , v = 13 , & w = 15 Hence x = 2, y = 3 & z = 5

Chapter 4 Class 12 Determinants

Ex 4.1, 7
Important

Example 14 Important

Example 15 Important

Example 16 Important

Ex 4.2, 7 Important

Ex 4.2, 8 Important

Ex 4.2, 11 Important

Ex 4.2, 12 Important

Ex 4.2, 13 Important

Ex 4.2, 14 Important

Ex 4.2, 15 Important

Example 18 Important

Ex 4.3, 2 Important

Ex 4.3, 3 Important

Example 24 Important

Example 26 Important

Ex 4.5, 10 Important

Ex 4.5, 15 Important

Ex 4.5, 18 Important

Ex 4.6, 13 Important

Ex 4.6, 15 Important

Ex 4.6, 16 Important

Example 32 Important

Example 34 Important

Misc. 2 Important

Misc 11 Important

Misc. 15 Important

Misc. 16 Important You are here

Misc 19 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.