(a)
(b)
19. Let D be an n by n determinant with cij the entry in the ith row and jth column. Show that
D = 0 if n is odd and cij + cji = 0 for all i and j.
20. Evaluate the n by n determinant with the entry cij in the ith row and jth column satisfying each of the following conditions: (It may be helpful to begin with small values of n and to try to find a pattern which suggests a proof.)
(a)
(b) cij = c1j if i > j.
(c) cij = a + |i -j|d. (See Section 8.4 for a definition of |x|.)
(d) cij = 1 if j - i is -1, 0, or a positive even integer, and cij = 0 for other values of j - i.
(e) cij = a + x if j > i, cij = b + x if j < i, and cii = ri + x.
(f)
