Assuming matrix $B$ is symmetric, can I prove that $A$ is symmetric

Question

$A,B$ are square matrices and $A(I+B)=I$, $B$ is symmetric, can I prove that $A$ is symmetric as well?

Answer 1

First note that $A(I+B)=I=(A(I+B))^T=(I^T+B^T)A^T=(I+B)A^T$. First right multiply $A(I+B)=I$ by $A^T$ to get $$A(I+B)A^T=IA^T=A^T$$ Then left multiply $(I+B)A^T$ by $A$ to yield $$A(I+B)A=AI=A$$ Thus $A^T=A$ and we conclude that $A$ is symmetric.

Answer 2

Given $B$ is symmetric.

$A(I+B) = I =(I+B)^T \times A^T = (I+B) A^T$ $$A(I+B)A^T = A = IA^T =A^T$$ so $A$ is symmetric!