Take the 2-minute tour ×  enter image description here  we need
to tell whether $a,b$ true or false,  I know that there does not
exist $n\times n$ $A,B$ such that $(AB-BA)=I$,as if we take
trace of both side they are not equal, so $a$ is false?  $b$ I
have no idea, could any one give me hint?  share|improve this
question if $(a)$ must be true for all $n$, the $(a)$ is false
(for $n=1$, it's impossible). –  Xoff Aug 14 '12 at 20:47 add
comment  2 Answers  up vote 3 down vote accepted  (a) Let $C =
AB - BA$. Then we have that the eigenvalues of $(I-C)^n$ are all
$0$, and hence the eigenvalues of $I - C$ are also all zero.
(Here we use the fact that if $\lambda$ is an eigenvalue of $A$
then $\lambda^n$ is an eigenvalue of $A^n$).  Hence, the
eigenvalues of $C$ are all $1$, which implies that the trace of
$C$ is $n$. On the other hand, $tr(AB) = tr(BA)$ implies that
$tr(C) = tr(AB - BA) = 0$, a contradiction.  So, no such
matrices exist.  (b) Take $a_1,a_2, \cdots, a_n$ to be the
eigenvalues of $A$, and note that