Find the canonical representative of the matrix equivalence class of the matrix $A$.

Question

Find the canonical representative of the matrix equivalence class of the matrix $A$.

I just have one question regarding this problem. What I by far understood is that equivalence class of the matrix $A$ consists of all those matrices which are equivalent to $A$. So if if reduce $A$ to a row-reduced form or a normal form then that resultant matrix will be the canonical representative of this class.

Am I correct??

P.S. I didn't write down what matrix $A$ because it's not relevant to my query. And also there was no other information in the question.

Answer 1

Theorem. If $A\in\mathbb{F}^{m\times n}$ and $\text{rank}(A)=r$ then, $A\sim\begin{bmatrix}I_r & 0\\ 0 & 0\end{bmatrix}\in\mathbb{F}^{m\times n}.$