Question

let A and B be two symmetric matrices .prove that AB = BA if and only if ABis a symmetric matrix. ​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A and B are two symmetric matrices}$

$\underline{\textbf{To prove:}}$

$\textsf{AB=BA if and only if AB is a symmetric matrix}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Concet used:}}$

$\textsf{(i) A square matrix A is said to be}\;\textbf{Symmetric}\;\mathsf{if\;A^T=A}$

$\textsf{(ii) If A and B are two square matrices of same order,}$

$\mathsf{then,\;\;(AB)^T=B^TA^T}$

$\textbf{Proof:}$

$\textsf{Suppose A and B are symmetric matrices}$

$\mathsf{Then,\;A^T=A\;\;\&\;\;B^T=B}$

$\mathsf{Consider,}$

$\mathsf{(A\,B)^T}$

$\mathsf{=B^T\,A^T}$

$\mathsf{=B\,A}$

$\mathsf{=A\,B}$

$\implies\mathsf{(A\,B)^T=A\,B}$

$\textsf{Hence, AB is symmetric matrix}$

$\textbf{Converse Part:}$

$\textsf{Suppose AB is a symmetric matrix}$

$\mathsf{Then,\;\;(AB)^T=AB}$

$\implies\mathsf{B^TA^T=AB}$

$\implies\mathsf{BA=AB}$

$\mathsf{Hence,\;AB=BA}$