-- Here are some sundry exercises.
-- HARD EXERCISES (ones I don't know how to answer myself).
-- Suppose you have a quick way of computing the determinant of a matrix
-- with entries in Z. How best to compute determinant of a matrix over Q?
-- One approach is to clear denominators in the matrix entries by
-- multiplying rows/columns by suitable integers. Alternatively, you could
-- try predicting a small multiple of the denominator of the determinant.
-- Consider a matrix whose first row and column are filled with 1/N,
-- and all other entries are integer. Another case to consider is
-- the matrix whose (i,j) entry is 1/(i*j).
-- Given a matrix M (over a ring R) and a value D in R, is there is quick
-- way of verifying whether D = det(M)? If so, probabilistic modular
-- methods can be used for computing determinants.
-- Is there an easy way to get a good upper bound on degree of
-- determinant of a matrix in R[x]? What about R[x_1,...,x_n]?