Ex 4.6, 5 - Examine consistency - Chapter 4 Determinants - Checking consistency of equations

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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise

Transcript

Ex 4.6, 5 Examine the consistency of the system of equations. 3x − y − 2z = 2 2y − z = −1 3x − 5y = 3 The system of equations can be written as 3x − y − 2z = 2 0x + 2y − z = −1 3x − 5y + 0z = 3 Step1 Write equation as AX = B 3﷮−1﷮−2﷮0﷮2﷮−1﷮3﷮−5﷮0﷯﷯ 𝑥﷮𝑦﷮𝑧﷯﷯ = 2﷮1﷮3﷯﷯ Hence A = 3﷮−1﷮−2﷮0﷮2﷮−1﷮3﷮−5﷮0﷯﷯, X = 𝑥﷮𝑦﷮𝑧﷯﷯ & B = 2﷮1﷮3﷯﷯ Step 2 Calculate |A| |A| = 3﷮−1﷮−2﷮0﷮2﷮−1﷮3﷮−5﷮0﷯﷯ = 3 2﷮−1﷮−5﷮0﷯﷯ – ( – 1) 0﷮−1﷮3﷮2﷯﷯ + 2 0﷮2﷮3﷮−5﷯﷯ = 3(0 – 5) + 1 (0 + 3) – 2 (0 – 6) = 3 ( – 5) + 1 (3) – 2 ( – 6) = – 15 + 3 + 12 = – 15 + 15 = 0 Since |A| = 0, We calculate adj A (B) adj A = A﷮11﷯﷮ A﷮12﷯﷮ A﷮13﷯﷮ A﷮21﷯﷮ A﷮22﷯﷮ A﷮23﷯﷮ A﷮31﷯﷮ A﷮32﷯﷮ A﷮33﷯﷯﷯﷮′﷯= A﷮11﷯﷮ A﷮21﷯﷮ A﷮31﷯﷮ A﷮12﷯﷮ A﷮22﷯﷮ A﷮32﷯﷮ A﷮13﷯﷮ A﷮32﷯﷮ A﷮33﷯﷯﷯ A = 3﷮−1﷮−2﷮0﷮2﷮−1﷮3﷮−5﷮0﷯﷯ M11 = 2﷮−1﷮−5﷮0﷯﷯ = 0 – (5) = – 5 M12 = 0﷮−1﷮3﷮0﷯﷯ = 0 + 3 = –3 M13 = 6﷮2﷮3﷮−5﷯﷯ = 0 – 6 = – 6 M21 = −1﷮−2﷮−5﷮0﷯﷯ = 0 – 10 = – 10 M22 = 3﷮−2﷮3﷮0﷯﷯ = 0 + 6 = 6 M23 = 3﷮−1﷮3﷮−5﷯﷯ = – 15 +3 = – 12 M31 = −1﷮−2﷮2﷮−1﷯﷯ = 1 + 4 = 5 M32 = 3﷮−2﷮0﷮−1﷯﷯ = – 3 + 0 = – 3 M33 = 3﷮−1﷮0﷮2﷯﷯ = 6 + 0 = 6 A11 = ( – 1)1+1 . M11 = ( – 1)2 . ( – 5) = – 5 A12 = ( – 1)1+2 . M12 = ( – 1)3 . 3 = – 3 A13 = ( – 1)1+3 . M13 = ( – 1)4 . ( – 6) = – 6 A21 = ( – 1)2+1 . M21 = ( – 1)3 . ( – 10) = 10 A22 = ( – 1)2+2 . M22 = ( – 1)4 . 6 = 6 A23 = ( – 1)2+3 . M23 = ( – 1)5 . ( – 12) = 12 A31 = ( – 1)3+1 . M31 = ( – 1)4 . 5 = 5 A32 = ( – 1)3+2 . M32 = ( – 1)5 . ( – 3) = 3 A33 = ( – 1)3+3 . M33 = ( – 1)6 . (6) = 6 Thus, adj (A) = A﷮11﷯﷮ A﷮21﷯﷮ A﷮31﷯﷮ A﷮12﷯﷮ A﷮22﷯﷮ A﷮32﷯﷮ A﷮13﷯﷮ A﷮32﷯﷮ A﷮33﷯﷯﷯ = 3﷮−1﷮−2﷮0﷮2﷮−1﷮3﷮−5﷮0﷯﷯ Now, adj (A) . B Putting values = −5﷮10﷮5﷮−3﷮6﷮3﷮−6﷮12﷮6﷯﷯ 2﷮−1﷮3﷯﷯ = −5 2﷯+10 −1﷯+5 3﷯﷮−3 2﷯+6 −1﷯+3 3﷯﷮−6 2﷯+12 −1﷯+6 3﷯﷯﷯ = −10−10+15﷮−6−6+9﷮−12−12+18﷯﷯ = −5﷮−3﷮−6﷯﷯ Thus, adj A . B ≠ O Therefore, |A| = 0 & (adj A) B ≠ O, Thus, the given system equation is inconsistent & the system of equations has no solution

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.