# linear algebra – Eigenvalues of scaling of matrix

Let $$A$$ be a (real or complex) square matrix, let $$alpha neq 0$$.

Is it true that $$lambda$$ is an eigenvalue of $$A$$ if and only if and only if $$alpha lambda$$ is an eigenvalue of $$alpha A$$?

I think yes, here is why I suppose so: $$lambda$$ is an eigenvalue of $$A ifflambda I- A$$ is non injective $$iff alpha lambda I – alpha A$$ is non injective $$iff alpha lambda$$ is an eigenvalue of $$alpha A$$.

Is this correct?