Prove that $A^T + A$ is symmetric for any $n \times n$ matrix $A$. So I understand how matrix Transpose works but I'm not sure if there is an example I can use to prove that this works for ANY $n \times n$ matrix.
Consider the following identity:
Now apply it: $$ (A^T+A)^T=(A^T)^T+A^T=A+A^T.$$ A matrix is symmetric if it equals its transpose. Matrix addition is commutative, and so: $$ A^T+A=A+A^T=(A^T+A)^T.$$ Thus, $A^T+A$ is a symmetric matrix for any $n\times n$ matrix $A$.
A square matrix $M$ is symmetric if (by definition) it coincides with its transpose, i.e. if $M^T=M$.
Now, if $M=A^T+A$, we have that $$ M^T=(A^T+A)^T=(A^T)^T+A^T=A+A^T=M\;\;\;\;, $$ where the second equality above, follows by linearity of the operator of transposition, and the third one follows because such operator is involutory.