How to prove: if $AB=I$ then $\mathrm{rank}(A)=\mathrm{rank}(B)$ and if $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B)\le n$
13026
Prove that if AB=I_m, then rk(A)=rk(B).
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
89175
Proof of $(a+b)^{n+1}$
33451
Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$
74611
$\operatorname{Adj} (\mathbf I_n x-\mathbf A)$ when $\operatorname{rank}(\mathbf A)\le n-2$
80009
Prove that $| |a|-|b| | \le |a-b|$
2798
A and B are nxn matrices, $A =B^TB$. Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.
104653
$A$ and $B$ are $3\times 3$ real matrices such that $\operatorname{rank}(AB)=1$, then $\operatorname{rank}(BA$) can not be which of the following?
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
78575
If $A^2 = I$, then $\operatorname{rank} (A+I) + \operatorname{rank} (A-I) =n$
65062
Prove $\text{rank}(A) \geq \frac{(\text{tr}(A))^2}{\text{tr}(A^2)}$ when $A$ is Hermitian
3301
If $AB = I$ then $BA = I$
121157
Proof verification: column rank = row rank
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
146847
Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
138707
show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$.