Can we make a matrix with these properties?

Question

1. $\dim N(A) = 1 + \dim N(A^{T})$

A=$\begin{bmatrix} 1& 1& 1\\ 0& 1& 1 \end{bmatrix}$.

Here $\dim N(A)=1$, and $\dim N(A^{T})=0$, so that matrices exist.I just want to be sure that I am right because I have very strict professor.

2) $R(A)=R(A^{T})$ and $N(A)\not= N(A^{T})$.

Since $R(A)$ and $R(A^{T})$ are subspaces of two different spaces and because $R(A)=R(A^{T})$, that means that dimension of matrices must be $n*n$, I know $\dim N(A)+\dim R(A)=n$ and $\dim N(A^{T})+ \dim R(A^{T})=n$. From here we see that $N(A)=N(A^{T})$ so that matrix does not exist

Answer 1

Yes. You are correct in both your answers and their reasoning