Example 27 - If A, B are symmetric matrices of same order, show - Examples

  1. Chapter 3 Class 12 Matrices
  2. Serial order wise
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Example 27, If A and B are symmetric matrixes of the same order, then show that AB is symmetric if and only if A and B commute, that is AB = BA. Given A & B are symmetric matrix i.e. A = A B = B We need to show AB is symmetric if and only if A & B commute (i.e. AB = BA) i.e. we need to show If AB is symmetric, then A & B commute (i.e. AB = BA) and If A & B commute (i.e. AB = BA), then AB is symmetric Proving Forward part If AB is symmetric then A & B commute Given AB is symmetric i.e. (AB) = AB B A = AB BA = AB Hence A & B commute. Hence proved Proving backward part If A & B commute, then AB is symmetric Given A & B commute i.e. AB = BA We need to show AB is symmetric i.e. we need to show (AB) = AB Taking (AB) = B A = BA = AB So, (AB) = AB Hence AB is symmetric Hence proved Hence, AB is symmetric if and only if A and B commute, that is AB = BA.

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Davneet Singh
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.