Proving $ab$ and $ba$ have the same order
98331
Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
138707
show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$.
78736
Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
59397
Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are square matrices
158695
Prove that if $AB$ is invertible then $B$ is invertible.
146795
Proof If $AB-I$ Invertible then $BA-I$ invertible.
62474
If $AB=I_n$ and $BA=I_m$ then $n=m$.
90475
Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?
6410
Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
6096
If $A+B=AB$ then $AB=BA$
21807
on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$
142980
If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$
113065
Proving equivalences of statements equivalent to AC
6091
Similarity of products $AB$ and $BA$ of square matrices
140290
Solving $AB - BA = C$
110502
$AB=BA$ with same eigenvector matrix