P.S. I've got not very elegant solution
there's also an elementary solution which doesn't use any linear algebra concept. we clearly have these identities:
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
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)?
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 .