Prove that if matrix $A$ is an $m\times n$ and $B$ is $n\times p$, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$
41026
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
115582
Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
59396
Show that the rank of $A+B$ is no more than the sum of the ranks of $A$ and $B$
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
75117
$\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable
94639
Proof that the rank of a skew-symmetric matrix is at least $2$
45469
If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
76200
Is the rank of $AB$ always equal to the rank of $BA$?
33451
Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$
10483
Prove Sylvester rank inequality: $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n$
74611
$\operatorname{Adj} (\mathbf I_n x-\mathbf A)$ when $\operatorname{rank}(\mathbf A)\le n-2$
69606
How to prove rank(A+) is no more than rank(A)?
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
111951
Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
13026
Prove that if AB=I_m, then rk(A)=rk(B).
116215
Prove that $P(A \cap B\mid C)=P(A \mid (B \cap C))P(B\mid C)$