Check sibling questions

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

Example 27 - Chapter 3 Class 12 Matrices - Part 2
Example 27 - Chapter 3 Class 12 Matrices - Part 3
Example 27 - Chapter 3 Class 12 Matrices - Part 4

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

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 (Using property (AB)’ = B’A’) (Given A’ = A & B’ = B) 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 (As (AB)’ = B’A’) (As A = A’ & B= B’ given) (Assumed that AB = BA) So, (AB)’ = AB Hence, AB is symmetric Hence proved Therefore, AB is symmetric if and only if A and B commute, i.e. AB = BA.

Davneet Singh's photo - Co-founder, Teachoo

Made by

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.