# Example 27 - Chapter 3 Class 12 Matrices

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

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 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.

Proof using property of transpose

Chapter 3 Class 12 Matrices

Concept wise

- Formation and order of matrix
- Types of matrices
- Equal matrices
- Addition/ subtraction of matrices
- Statement questions - Addition/Subtraction of matrices
- Multiplication of matrices
- Statement questions - Multiplication of matrices
- Solving Equation
- Finding unknown - Element
- Finding unknown - Matrice
- Transpose of a matrix
- Symmetric and skew symmetric matrices
- Proof using property of transpose
- Inverse of matrix using elementary transformation
- Proof using mathematical induction

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.