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 $\operatorname{rank} B$
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
69980
Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
75211
Theorem about positive matrices
21758
Rank of the $n \times n$ matrix with ones on the main diagonal and $a$ off the main diagonal
48660
Proof that determinant rank equals row/column rank
55820
If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
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?
48658
Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is similar to $B$. Prove trace($A$) = trace($B$).
128760
If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
27474
A square matrix $n \times n$ is an invertible matrix iff the rank of the matrix is $n$.
146847
Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
153823
$|\text{rank}A-\text{rank}B|\leq \text{rank}(A+B)$
50266
In the following figure prove that: $AK||BC$
158845
Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$.
102817
$A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
33451
Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$