$T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined as $T(x,y,z) = (2x,z,y)$ is a linear transformation. 

I need to prove that the following matrices cannot represent $T$ in ANY basis:

$$\begin{bmatrix} 2 & 0 & 1 \\
0 & 1 & 0\\
2 & 0 & 1
\end{bmatrix}$$

$$\begin{bmatrix} 1 & 0 & 1 \\
1 & 0 & 0 \\
0 & 2 & 1\end{bmatrix}$$

My attempt for the first matrix was to assume (negatively) existence of a basis in which the given matrix is the representation, and left multiplying it by $(x,y,z)^{T}$ to get $(2x+z,y,2x+z)^{T}$. 

Can I conclude that given it is not in the form $(2x,z,y)^{T}$, there is NO such basis? 

I want to know if this is correct and, in addition (Or alternatively), learn other ways to solve this exercise. 

Thank you.