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Matrices and Determinants - Formula Sheet and Summary Important

Chapter 4 Class 12 Determinants

Serial order wise

Last updated at March 30, 2023 by Teachoo

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