Exercise 2. Multiplying Special Matrices

a. What happens when you multiply an upper(lower) triangular matrix with another upper(lower) triangular matrix?

b. What happens when you multiply a matrix by a diagonal matrix with the diagonal matrix on the right. With it on the left?

c. What happens when you raise a diagonal matrix to a power?

d. A matrix is nilpotent of order k (integer) if A^k=the zero matrix and A^k is not 0 for any integer less than k. Consider matrices with 1's on the superdiagonal and 0's everywhere else. First verify that these matrices are nilpotent and then find a relationship between the size of the superdiagonal matrix and the order of nilpotency.

e. Write out a general formula for A^k when A={{1,1},{0,1}}.

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