Comparing the rank of the product of two matrices with the ranks of the factors Assume matrices $A_{m\times n }$ and $B_{n\times p}$


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*Show that Rank $(AB)\leq $ Rank $(A)$
What i tried is this:
Col (AB) = {(AB)x : x in R^p} = {A(Bx) : x in R^p}
But (Bx) is in R^n, so it's clear that if y is in A(Bx), then y is in Ax                  (here, x is in R^n)
So Col(AB) is a subspace of Col(A) and thus dim Col(AB) <= dim Col(A)
I am not sure if I am right.

*Show that Rank $(AB)\leq $ Rank $(B)$
Rank$(AB)^T$ = rank ($B^T A^T$) $\leq$ Rank $B^T$ by part (1)
But Rank $B^T$ is the dimensions of the null space of B. So,
Rank $B^T$ = dim Nul B = n - rank B

*Show that if P is an invertible m x m matrix, then Rank (PA) = Rank A

*Show that if Q is invertible, then Rank (AQ) = Rank A

*Show that if A and B are similar, then they have the same Rank.
The professor said that we can use the previous proofs to show the next one.
Any help please.
 A: $(1):$ Note that $\text{Im}(AB)\subset\text{Im}(A)$.
$(2):$ Note that $\ker(B)\subset\ker(AB)$ and use the rank-nullity theorem.
$(3):$ Observe that $\ker(PA)=\ker(A)$ and use the rank-nullity theorem.
$(4):$ Observe that $\text{Im}(AQ)=\text{Im}(A)$.
$(5):$ Write $B=P^{-1}AP$ and check that $\ker(B)=\ker(AP)$ -- since $P$ is invertible, $P(a)=0$ if and only if $a=0$. Then use $(4)$ to conclude that $A$ and $AP$ have the same rank (and hence also the same nullity). It follows that $\text{nul}(B)=\text{nul}(A)$. Finally, use the rank-nullity theorem.
A: Part (1) is just perfect: the rank of $A$ is the dimension of the image of the linear map associated to $A$. Now the image of the linear map associated to $AB$ (assuming column vectors) is a subset of the image of the linear map associated to $A$.
Part (2) is brilliant, honest, only, write $\operatorname{Rank}((A B)^{T})$.
As to part (3), note that $A = P^{-1} (P A)$ and use the previous points twice: you will get two inequalities which yield the required equality.
Same with (4).
As to (5), $A$ and $B$ similar means $B = P^{-1} A P$ for some invertible $P$. Use (3) and (4).
