Why does Det[Transpose[A]] == Det[A]?

You can get the transpose of a 2-d matrix by hitting it with a matrix like {{0, 1},{1, 0}}.

So Transpose[{{2, 4},{6, 8}}] == {{2, 4},{6, 8}} . {{0, 1},{1, 0}}. It has the effect of turning rows into columns.

If you calculated the determinant of that special matrix (call it S) of 1's and 0's, you'd find that it's determinant is always going to be 1.

Since Det[A . B] == Det[A] Det[B], you know that Det[Transpose[A]] == Det[A] Det[S] == Det[A]