A and B are similar if and only if $\text{rank}(A)=\text{rank}(B)$?
73297
Two idempotent matrices are similar iff they have the same rank
27767
Are two matrices similar iff they have the same Jordan Canonical form?
6091
Similarity of products $AB$ and $BA$ of square matrices
87387
Linear equations given by $Ax = b$ have a solution if and only if $\operatorname{rank}(A|b) = \operatorname{rank}(A)$
110661
When does $\sqrt{a b} = \sqrt{a} \sqrt{b}$?
6096
If $A+B=AB$ then $AB=BA$
139685
If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?
64517
Prove that row rank of a matrix equals column rank
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
79526
How do I tell if matrices are similar?
155545
Showing that $a \mid b$ and $b \mid a$ if and only if $a= \pm b$.
41044
Equivalence of $a \rightarrow b$ and $\lnot a \vee b$
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
133926
What if 2 answers are essentially the same?
17430
If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$
21229
If an element is not in the set difference $A\setminus B \: \:?$
110381
Prove that if rank(A + B) = rank(A) + rank(B), then col(A) ∩ col(B) = {0}