Define the indices that you want to interchange in a list $ind$ and then you can index into the array $a$ directly.
a = {{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}, {3, 4, 5, 6, 7}, {4, 5, 6, 7, 8}, {5, 6, 7, 8, 9}}; ind = {1, 3, 4, 2, 5}; a[[ind, ind]] If you don't want to specify the index array each time, Roman points out that you can build a simple function to do it:
indexlist[n_, i_, j_] := Insert[Delete[Range[n], i], i, j] SOSo to get the above you would specify
ind = indexlist[5,2,4]