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Last updated at Dec. 8, 2016 by Teachoo

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Misc 18 Choose the correct answer. If x, y, z are nonzero real numbers, then the inverse of matrix A = x000y000z is A. 𝑥−1000 𝑦−1000 𝑧−1 B. xyz 𝑥−1000 𝑦−1000 𝑧−1 C. 1xyz x000y000z D. 1xyz 100010001 Given A = x000y000z We have to find A-1 We know that A-1 = 1|A| adj (A) exists if |A| ≠ 0 Calculating |A| |A| = x000y000z = x 𝑦00𝑧 – 0 000𝑧 + 0 0𝑦00 = x (yz – 0) – 0 (0 – 0) + 0 (0 – 0) = x(yz) + 0 + 0 = xyz Since |A| ≠ 0 Thus, A-1 exist Now, adj A = A11 A21 A31 A12 A22 A32 A13 A23 A33 A = 𝑥000𝑦000𝑧 M11 = 𝑦00𝑧 = yz – 0 = yz M12 = 000𝑧 = 0 – 0 = 0 M13 = 0y00 = 0 – 0 = – 0 M21 = 000𝑧 = 0 – 0 = 0 M22 = x00𝑧 = xz – 0 = xz A11 = ( – 1)1 + 1 M11 = ( – 1)2 y z = yz A12 = ( – 1)1+2 M12 = ( – 1)3 0 = 0 A13 = ( – 1)1+3 M13 = ( – 1)4 0 = – 0 A21 = ( – 1)2+1 M21 = ( – 1)3. 0 = 0 A22 = ( – 1)2+2 M22 = ( – 1)4 x z = x z A23 = ( – 1)2+3 M23 = ( – 1)5 0 = 0 A31 = ( – 1)3+1 M31 = ( – 1)4 0 = 0 A32 = ( – 1)3+2 M32 = ( – 1)5 0 = 0 A33 = ( – 1)3+3 M33 = ( – 1)6 xy = xy Thus, adj (A) = A11A21A31A12A22A32A33A23A33 = 𝑥 𝑦000𝑧 𝑦000𝑥 𝑦 Now, A-1 = 1|A| adj (A) = 1𝑥𝑦𝑧 𝑥 𝑦000𝑧 𝑦000𝑥 𝑦 = 𝑦𝑧𝑥𝑦𝑧000 𝑦𝑧𝑥𝑦𝑧000 𝑦𝑧𝑥𝑦𝑧 = 1𝑥000 1𝑦000 1𝑧 = 𝑥−1000 𝑦−1000 𝑧−1 Thus, the correct option is A

Miscellaneous

Misc 1

Misc. 2 Important

Misc 3

Misc 4

Misc 5

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11 Important

Misc 12 Important

Misc. 13

Misc 14

Misc. 15 Important

Misc. 16 Important

Misc 17 Important

Misc 18 You are here

Misc 19 Important

Matrices and Determinants - Formula Sheet and Summary

Chapter 4 Class 12 Determinants

Serial order wise

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.