Question 6.2.2: Consider the matrices A[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 ......
Consider the matrices
Note that B is obtained from A by swapping the first two rows. Express det B in terms of det A.
For each pattern P in A, we can consider the corresponding pattern P_{swap} in B; for example,
These two patterns P and P_{swap} involve the same numbers, but the number of inversions in P_{swap} is one less than in P, since we are losing the inversion formed by the entries in the first two rows of A. Thus, prod P_{swap} = prod P, but sgn P_{swap} = −sgn P, so that the two patterns make opposite contributions to the respective determinants. Since these remarks apply to all patterns in A, we can conclude that
\det B=-\det A.
(If P is a pattern in A such that the entries in the first two rows do not form an inversion, then an additional inversion is created in P_{swap}; again, sgn P_{swap} = −sgn P.)