I have this exercise but I don't have acquired yet all the information I need in order to deal with it, at least.
Show that if an upper (lower) triangular matrix has only zeros on the main diagonal, then there is some n ∈ N such that the n-th power of the matrix is zero. Can such a matrix (and the corresponding linerar operator) be invertible (why, or why not)?.
As far as I know, in order for a triangular matrix to be invertible, can't be any zero in its diagonal. More in general, I know that when a matrix has an inverse, the product of the matrix and its inverse is equal to the identity matrix - I know that the identity matrix is the one with 1s as its diagonal. Thus, I know the answer to the last question in bold character. Namely, I think that a matrix with 0s as its diagonal is not invertible. Why? I can only give the reasons above as still I am not confident with proof-writing.
However, the task also asks to show that if an upper (lower) triangular matrix has only zeros on the main diagonal, then there is some n ∈ N such that the n-th power of the matrix is zero. How can I prove this?