taking complex matrices makes it easier to understand. any complex matrix is similar to a upper traingular one. so wlog u may assume that u started with an upper traingular one (why?).

any power of an upper traingular matrix is also upper traingular... so all the powers will at most span only upper traingular matrices, leaving out the rest.

for real matrices you may argue the same way using jordan forms. in this case too the answer seems to be 'no'.