Question on Rank of a matrix

Question

Let $A$ be an $n \times n$ matrix with $\operatorname{rank}(A) = k$, what is the rank of ($\lambda A$) for some $\lambda \in \mathbb{R}$ ?

I think since $\lambda$ is just a scalar it won't affect the rank of the matrix, so the answer is still $k$ ?

Answer 1

Good guess so far.

However, don't forget the special case where $\lambda =0$, of which case, the problem become what is the rank of $0\cdot A=0$.

When $\lambda \neq 0$, solution of $(\lambda A)x=0$ and the solution of $Ax=0$ is the same. That is the kernel is the same. By rank-nullity theorem, the rank remains the same, which is $k$.