question_answer

Show that all the positive integral powers of a symmetric matrix are symmetric.

Answer:

Let A be a symmetric matrix and \[n\in N.\] Then, \[{{A}^{n}}=AAA...A\] upto n-times \[\Rightarrow \]   \[{{({{A}^{n}})}^{T}}={{(AAA...A\,\,\text{upto}\,\,n\text{-times)}}^{T}}\] \[\Rightarrow \]   \[{{({{A}^{n}})}^{T}}=({{A}^{T}}{{A}^{T}}{{A}^{T}}...{{A}^{T}}\,\,\text{upto}\,\,n\text{-times)}\] \[\Rightarrow \]   \[{{({{A}^{n}})}^{T}}={{({{A}^{T}})}^{n}}={{A}^{n}}\]          \[[\because \,\,{{A}^{T}}=A]\] Hence, \[{{A}^{n}}\] is also a symmetric matrix.