Question 6.2.1: Let A=[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9]. E......

For each pattern P in A, we can consider the corresponding (transposed) pattern P^T in A^T ; for example,

The two patterns P and P^T involve the same numbers, and they contain the same number of inversions, but the role of the two numbers in each inversion is reversed. Therefore, the two patterns make the same contributions to the respective determinants (sgn P)(prod P) = (sgn P^T )(prod P^T ). Since these observations apply to all patterns of A, we can conclude that det(A^T ) = det A.