If we have a square matrix thats invertible, do its row and column space coincide?

Question

If we have a square matrix thats invertible, do its row and column space coincide?

Regarding an nxn invertible matrix:

-The row space of the matrix is R^n

-The column space of the matrix is R^n

-The rank of the matrix is n

Is this a sufficient way of proving the question, or am I missing something?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$The row space and column space of an $n \times n$ matrix are not generally equal, e.g., $$ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad \text{row space} = \{0\} \times \Reals,\quad \text{column space} = \Reals \times \{0\}. $$ The row space and column space of an $n \times n$ matrix do always have the same dimension, however, and if this dimension is $n$, then each space is equal to $\Reals^{n}$.