there's also an elementary solution which doesn't use any linear algebra concept. we clearly have these identities:
i)
ii)
now let be the identity matrix and let then note that and hence, using binomial theorem, we have:
thus by i) and ii) we have:
now and therefore
that is correct! another application is finding powers of some low dimensional upper (lower) triangular matrices. for example if then where
now, look at and then which is zero. can you find a general fact about powers of strictly triangular matrices (that is triangular matrices with all diagonal entries equal to 0)?
anyway, so
It is a tricky thing indeed , we have must be zero for is matrix that you mentioned here , because at first the ' length ' of the adjacent side of the right-angled isosceles triangle is , when we multiply it by , the 'length' decreases by 1 unit so after we multiply times , the length becomes zero or say . I think we just need to find out .