I am a little stuck on this Matrix problem. Suppose that for complex square matrix A,B the following holds:
$AB -BA = A$ Show that $\det(A)=0$
That would mean that A has no inverse. So I thought, let's suppose there exists an Inverse element and that would lead me to a contradiction later on.
$AB = A + BA \implies AB = A(I+B)$ multiply though with $(AB)^{-1}$
$I = B^{-1}A^{-1}A(I+B)$
$I=B^{-1}(I+B)$
$I=B^{-1} +I$ so $B^{-1}$ is equal to $0$. Contradiction in my view. Or what does this mean? Could you please help me interpret? :)
Thanks in advance!