Let D be as in (1.45).b) Let aij = 1 if i + j = n + 1 and aij = 0 otherwise. Show that D =...

Let D be as in (1.45).

b) Let aij = 1 if i + j = n + 1 and aij = 0 otherwise. Show that D = −1.

c) Let aij = 0 for i > j, so that the array “upper triangular form.” Show that D = a11a22…ann.

d) It can be shown that

where the sum is over all permutations (j1…jn) of (1, 2,..., n) and ɛj1j2…jn is 1 for a permutation which is even (obtainable from (1, 2,..., n) by an even number of interchanges of two integers) and is − 1 for an odd permutation (odd number of interchanges). (See Chapter 4 of the book by Perlis listed at the end of the chapter.) Verify this rule for n = 2 and for n = 3.