How to find the range of $T : M_n(\Bbb{R}) \to M_n(\Bbb{R})$ given by $T(A) = A + A^t$?

Question

How to find the range of the linear transformation given by $$T(A)=A+A^t$$ with $A$ being an $n\times n$ matrix?

I have absolutely nothing except that $(A+A^t)^t=A+A^t$ which would make $A+A^t$ symmetric but how does that affect the range?

Answer 1

For a square matrix $A\in\mathcal M_n(\Bbb R)$ it's clear that $T(A)$ is a symmetric matrix so $$\operatorname{im} T\subset \mathcal S_n(\Bbb R)$$ and if $B\in \mathcal S_n(\Bbb R)$ then $T\left(\frac12 B\right)=B$ hence $$\mathcal S_n(\Bbb R)\subset \operatorname{im} T$$ so we conclude that $$\mathcal S_n(\Bbb R)=\operatorname{im} T$$