Given $n \times n$ matrices, $A$ and $B$, show that ${\rm rank}(AB) \leq {\rm rank}(A)$
41026
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
25488
Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
34536
Columns of matrix $A_{m \times n}$ span $\mathbb{R}^{m}$
151443
Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,...but they are not similar.
45532
Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$
27821
A problem on rank of $A$ and $\operatorname{adj}(A)$
48658
Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is similar to $B$. Prove trace($A$) = trace($B$).
134402
$\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
51800
We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$
111031
Matrix rank and linear independence of functions
54740
Proofs of Determinants of Block matrices
55820
If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
257
Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
43000
Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$
82295
Showing that the minimal polynomial of an $n \times n$ matrix has degree at most $n$ without using the Cayley-Hamilton Theorem
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
9574
How to show that $\det(AB) =\det(A) \det(B)$?