(a) Consider some permutation of the integers 1, 2, 3, ( ( ( ( n. The permutation is an even permutation if an even number of interchanges of pairs of integers restores the permutation to the natural order 1, 2, 3, ( ( ( ( n. An odd permutation requires an odd number of interchanges of pairs to reach the natural order. For example, the permutation 3124 is even, since two interchanges restore it to the natural order: 3124 ( 1324 ( 1234. Write down and classify (even or odd) all permutations of 123.
(b) Verify that the definition (8.24) of the third-order determinant is equivalent to

Where ijk is one of the permutations of the integers 123, the sum is over the 3! different permutations of these integers, and the sign of each term is plus or minus, depending on whether the permutation is even or odd.
(c) How would we define the nth-order determinant using this type of definition?

Transcribed Image Text:

aii a12 a13 a21 a22 a23_ Σ(±l)ali a2j@g azi d32 a33