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Ex 4.6, 16 - Cost of 4 kg onion, 3 kg wheat, 2 kg rice is Rs 60

Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 2
Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 3
Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 4
Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 5
Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 6
Ex 4.6, 16 - Chapter 4 Class 12 Determinants - Part 7

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Transcript

Ex 4.6, 16 The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix methodLet the cost of onion, wheat & rice per kg be x, y, z respectively. Given, Cost of 4 kg onion, 3 kg wheat & 2 kg rice is Rs 60. 4x + 3y + 2z = 60 Cost of 2 kg onion, 4 kg wheat & 6 kg rice is Rs 90. 2x + 4y + 6z = 90 Cost of 6 kg onion, 2 kg wheat & 3 kg Rice is Rs 70 6x + 2y + 3z = 70 So the system of equation is 4x + 3y + 2z = 60 x + 2y + 3z = 45 6x + 2y + 3z = 70 Writing equation as AX = B [■8(4&3&2@1&2&3@6&2&3)][■8(𝑥@𝑦@𝑧)] = [■8(60@45@70)] Hence A = [■8(4&3&2@1&2&3@6&2&3)], X = [■8(𝑥@𝑦@𝑧)] & B = [■8(60@45@70)] Calculating |A| |𝐴| = [■8(4&3&2@1&2&3@6&2&3)] = 4 (6 − 6) − 1(9 − 4) + 6 (9 − 4) = 4 (0) −1 (5) + 6 (5) = −5 + 30 = 25 So, |A|≠ 0 ∴ 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 = [■8(A11&A12&A13@A21&A22&A23@A31&A32&A33)]^′ = [■8(A11&A21&A31@A12&A22&A32@A13&A23&A33)] A = [■8(4&3&2@1&2&3@6&2&3)] Now, 𝐴11 = 2 × 3 −2 × 3 = 6 − 6 = 0 𝐴12 = −[3 × 1−6 × 3] = −[3−18] = −(−15) = 15 𝐴13 = 1 × 2 − 6 × 2 = 2 − 12 = −10 𝐴21 = −[3 × 3−2 × 2 ] = − [9−4] = −5 𝐴22 = 4 × 3 − 6 × 2 = 12 − 12 = 0 𝐴23 = − [2 × 1−6 × 2] = −[2−12] = − (−10) = 10 𝐴31 = 3 × 3 − 2 × 2 = 9 − 4 = 5 𝐴32 = −[4 × 3−2 × 1] = [12−2]=−10 𝐴33 = 4 × 2 − 3 × 1 = 8 − 3 = 5 Hence, adj (A) = [■8(0&−5&5@15&0&−10@−10&10&5)] Now, A−1 = 1/|𝐴| adj (A) A−1 = 1/25 [■8(0&−5&5@15&0&−10@−10&10&5)] A−1 = 1/5 [■8(0&−1&1@3&0&−2@−2&2&1)] Now, X = A−1 B [■8(𝑥@𝑦@𝑧)] = 1/5 [−■8(0&−1&1@3&0&−2@−2&2&1)] [■8(60@45@70)] [■8(𝑥@𝑦@𝑧)] = 1/5 [█(−0−45+70@180+0−140@−120+90+70)] = 1/5 [■8(25@40@40)] [■8(𝑥@𝑦@𝑧)] = [■8(5@8@8)] ∴ x = 5, y = 8, z = 8 Cost of onion, wheat & rice per kg is Rs 5, Rs 8 & Rs 8 Respectively.

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.