Linear operator, when it will be surjection and when it would not be?

Question

Let $A\in M4$ and $p(\lambda)=\lambda^{3}(\lambda-1)$. Linear transformation $A:R^{4}\to R^{4}$ defined $A(x)=Ax$, for some matrices linear operator it is surjection, for some matrices is not.

Now if we want surjection linear operator than $Im(A)=R^{4}$ in this case we need to have $ker(A)=\{0v\}$, but here we have $\lambda=0$ so $dimKer(A)\geq1$ than i think that for any matrices this linear operator never be surjection, but what do you think?

Answer 1

Yes, this is correct. Another way to see it is that $$\det(A) = p_A(0) = 0\ ,$$ so that $A$ is not invertible.