How can I prove that if $\ trace(C)=0 $ then $C=AB-BA$?
35801
Traceless matrices and commutators
63535
Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
20659
Is trace invariant under cyclic permutation with rectangular matrices?
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
151607
Show that  $A\Delta B = C$ if and only if $A = B\Delta C$
3301
If $AB = I$ then $BA = I$
64108
how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?
56102
Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$
66709
Are the matrix products $AB$ and $BA$ similar?
94727
When the trace of the Frobenius homomorphism is zero?
49330
If for two suitable matrices $B$ and $C$, $AA^TB=AA^TC$ can we conclude that $A^TB=A^TC$?
23224
$\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$
72507
Does $ab=1$ imply $ba=1$ in a ring?
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
88349
Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)
84626
Why in these matrices are $AB=BA$ not equal? What is the logic behind them?
156712
Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$