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I need help with this proof relating to strictly upper-triangular

matrices.

Let A be an n x n strictly upper triangular matrix. Then the (i,j-th

entry of AA = A^2 is 0 if i >= j - 1.

Here's what I have.

Pf: Let B = A^2. The (i,j)-th entry of B is given by

b_{ij} = \sum_{k=1}^{n} a_{ik} a_{kj}.

If k >= j, a_{kj} = 0. If i >= k, a_{ik} = 0. If i >= j - 1, then there

is no k s.t. i < k < j. Therefore

b_ij = \sum_{i<k, k<j} a_{ik} b_{kj} ==> b_{ij} = 0.

Also, if anyone has any clues on these related problems, it would be

greatly appreciated.

Suppose p is a given integer satisfying 1 <= p <= n -1 and that the

entries b_{kj} of an n x n matrix B satisfy b_{kj} = 0 for k >= j - p.

Show that the (i,j)-th entry of the product AB is zero if i >= j -

(p+1). Deduce from the previous result that A^n = 0.

James