This question is asked in Topics of Algebra by I N Herstein –
In $\displaystyle F_{n}$ let $\displaystyle S$ denote the set of all symmetric matrices. Prove that the subring of $\displaystyle F_{n}$ generated by $\displaystyle S$ is all of $\displaystyle F_{n}$
$\displaystyle F_{n}$ is the set of all $\displaystyle n\times n$ matrices over $\displaystyle F$.
My solution: for all $\displaystyle A$ and $\displaystyle B$ in $\displaystyle S$, $\displaystyle AB$ will be in subring generated by $\displaystyle S$. But $\displaystyle AB$ need not to be symmetric as $\displaystyle AB$ will be symmetric if and only if \ $\displaystyle A$ and $\displaystyle B$ commute. This implies that Subring will contain elements that are not symmetric.
But does this imply that subring generated by $\displaystyle S$ will contain skew symmetric elements and will be all of $\displaystyle F_{n}$?
Best Answer
In $F_2$ we have $A=\pmatrix{1&1\\1&1}$ and $B=\pmatrix{1&0\\0&0}$ are both symmetric; $AB-BA$ is non-trivially skew symmetric. $F_n$ for $n>2$ works the same way.