Rank in Linear Systems

Question

For any linear system $Ax=b$, is it true that $\text{rank}(A|b)=\text{rank}(A)$ ?

It doesn't mention the dimension of both $A$ and $b$.

Answer 1

This is incorrect, in general. It is true only when $b$ is in the range or column space of the operator $A$, i.e., if the system $Ax=b$ has at least one solution. For instance, consider $$A = \begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix} \text{ and }b = \begin{bmatrix} 1\\ 2\end{bmatrix}$$ We have $\text{rank}(A) = 1$, whereas $\text{rank}(A \vert b) = 2$.