How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
41026
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
121264
Prove that rank $(g \circ f)\le \min[\operatorname{rank}(f),\operatorname{rank}(g)]$; $g$ and $f$ are linear mappings
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
31302
Hilbert Schmidt Norm-Rank-inequality
155884
What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
45469
If $a_k = \operatorname{rank}(S^k)$, deduce that $a_k - a_{k+1} \geq a_{k+1} - a_{k+2}$
70278
$A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ -- how to prove this?
51347
Prove that $|a|\leq \max\{|b|,|c|\}$ if $b\leq a \leq c$
122326
Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
59396
Show that the rank of $A+B$ is no more than the sum of the ranks of $A$ and $B$
13228
Proofs of AM-GM inequality
23264
Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $
76347
Shorter proof for some equvalences
74611
$\operatorname{Adj} (\mathbf I_n x-\mathbf A)$ when $\operatorname{rank}(\mathbf A)\le n-2$
63036
Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$
21758
Rank of the $n \times n$ matrix with ones on the main diagonal and $a$ off the main diagonal