Is this true that $\operatorname{rank}A=\operatorname{rank}(AA^*)$?
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
43000
Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$
41398
For a ring $R$, does $\operatorname{End}_R(R)\cong R^{\mathrm{op}}$?
135422
$\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A') \Rightarrow A \cong A'$, proof verification
117688
Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
74187
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
55820
If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
11895
If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
129732
What is $\operatorname{Aut}(\mathbb{Z}_2\oplus \mathbb{Z}_2)$?
25009
$\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
121767
Why $\det(A^{-1}) = 1/\det(A)$?
125348
Does $\det(A + B) = \det(A) + \det(B)$ hold?
30923
Is the matrix $A^*A$ and $AA^*$ hermitian?
117533
$|A-\lambda I|$ and $|\lambda I-A|$
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$
19610
$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$