Prove is true or not

Question

Let $A,B\in Mn(\mathbb R)$ and $0$ is not eigenvalue of matrix $B$. For every case, tell is true or not then prove.

$1)R(A)=R(AB)$

$2)R(A)=R(BA)$

$3)N(A)=N(AB)$

$4)N(A)=N(BA)$

First notice that $dim R(B)=n$, first is true, we get every vector from $imB$ and that mean we get every vector from $imB$ so that mean we get every vector from $\mathbb R^n$ will belong $kerA$ or $imA$ so it can not be $dimimAB<dimimA$ or $dimimAB>dimimA$, second is not true because $dimimB=n$ and $dimimA<n$ so it can not be $R(A)=R(BA)$, third is not true it mean that $dimimB<dimkerA$ it is not true, end fourth is true, what you think?

Answer 1

Hints

2) and 3) Consider $A=\begin{pmatrix}1& 1\\ 1 & 1\end{pmatrix}$ and $B=\begin{pmatrix}1& 0\\ 0 & -1\end{pmatrix}.$

4) $x\in N(A)\iff Ax=0\iff BAx=0\iff x\in N(BA).$

Answer 2

Is enough to prove 1) and 4) and for 2) and 3),which we know aren't true, to use just example and in that way to prove they are false?