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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Ex 4.5, 11 Solve system of linear equations, using matrix method. 2x + y + z = 1 x – 2y – z = 3/2 3y – 5z = 9 The system of equation is 2x + y + z = 1 x – 2y – z = 3/2 3y – 5z = 9 Writing above equation as AX = B [■8(2&1&1@1&−2&−1@0&3&−5)][■8(𝑥@𝑦@𝑧)] = [■8(1@3/2@9)] Hence A = [■8(2&1&1@1&−2&−1@0&3&−5)]𝑥= [■8(𝑥@𝑦@𝑧)] & B = [■8(1@3/2@9)] Calculating |A| |A| = |■8(2&1&1@1&−2&−1@0&3&−5)| = 2 |■8(−2&−1@3&−5)| – 1 |■8(1&−1@0&−5)| + 1 |■8(1&−2@0&3)| = 2 (10 + 3 ) – 1(–5 + 0) + 1 (3 – 0) = 2 (13) –1 ( – 5 ) + 1 (3) = 34 Thus, |A| ≠ 0 ∴ The system of equation is consistent & has a unique solution Now, AX = B X = A-1 B Calculating A-1 A-1 = 1/(|A|) adj (A) adj (A) = [■8(A_11&A_12&A_13@A_21&A_22&A_23@A_31&A_32&A_33 )]^′ = [■8(A_11&A_21&A_31@A_12&A_22&A_23@A_13&A_32&A_33 )] A = [■8(2&1&1@1&−2&−1@0&3&−5)] M11 = [■8(−2&−1@3&−5)] = 10 + 3 = 13 M12 = [■8(1&−1@0&−5)] = –5 + 0 = –5 M13 = [■8(1&−2@0&3)] = 3 + 0 = 3 M21 = [■8(1&1@3&−5)] = –5 – 3 = –8 M22 = [■8(2&1@0&−5)] = –10 + 0 = –10 M23 = [■8(2&1@0&3)] = 6 + 0 = 6 M31 = [■8(1&1@−2&−1)] = –1 + 2 = 1 M32 = [■8(2&1@1&−1)] = –2 – 1 = –3 M33 = [■8(2&1@1&−2)] = –4 – 1 = –5 Now, A11 = (–1)1+1 . M11 = (–1)2 . (13) = 13 A12 = (–1)1+2 . M12 = (–1)3 . (–5) = 5 A13 = (–1)1+3 . M13 = (–1)4 . (3) = 3 A21 = (–1)2+1 . M21 = (–1)3 . (–8) = 8 A22 = (–1)2+2 . M22 = (–1)4 . (–10) = –10 A23 = (–1)2+3 . M23 = (–1)5 . (6) = – 6 A31 = (–1)3+1 . M31 = (–1)4 . (1) = 1 A32 = (–1)3+2 . M32 = (–1)5 . (–3) = 3 A33 = (–1)3+3 . M33 = (–1)6 . (–5) = – 5 Thus, adj (A) =[■8(13&8&1@5&−10&3@3&−6&−5)] Now, A-1 = 1/(|A|) adj A Putting values = 1/34 [■8(13&8&1@5&−10&3@3&−6&−5)] Also, X = A-1 B Putting values [█(■8(𝑥@𝑦)@𝑧)] = 1/34 [■8(13&8&1@5&−10&3@3&−6&−5)][█(■8(1@3/2)@9)] [█(■8(𝑥@𝑦)@𝑧)] = 1/34 [■8(13(1)+8(3/2)+1(9)@5(1)+(−10)(3/2)+3(9)@3(1)+(−6)(3/2)+(−5)(9) )] [█(■8(𝑥@𝑦)@𝑧)] = 1/34 [■8(13+12+9@5−15+27@3−9−45)] = 1/34 [█(■8(34@17)@−51)] [█(■8(𝑥@𝑦)@𝑧)] = [█(■8(1@1/2)@(−3)/2)] Hence x = 1, y = 𝟏/𝟐 & z = (−𝟑)/𝟐

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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.