[Math] Must a matrix be symmetric if Row Space = Column Space

Question

I have a matrix which the Row Space=Column Space, and I need to prove that the matrix is symmetric.

I need your help please!

Answer 1

I don't think this is true. Consider the matrix $\begin{bmatrix}1&1\\0&1\end{bmatrix}$.

The row space and the column space is $\mathbb{R}^2$, but the matrix is not symmetric.