The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n.$. True or False $?$ [duplicate]

Can any matrix be represented as a finite sum of nilpotent matrices?

Of course, if the diagonal is filled with zeros it is possible. What about $I_n$, the identity matrix, for example?

Thanks.

Solution 1:

Nilpotent matrices have zero trace, and thus any linear combination of nilpotent matrices has zero trace.