Determinant of a real skew-symmetric matrix What will be the value of the determinant of a skew-symmetric matrix of even order when a single element is interchanged between first row and first column?
For,
$\left|  \begin{array}{cccc}
0  & 1  & 2 & -1 \\
-1 & 0  & 1 & 2  \\
-2 & -1 & 0 & 1  \\ 
1  & -2 & -1 & 0 \end{array} \right|$ = 16
$\left|  \begin{array}{cccc}
0  & 1  & -2 & -1 \\
-1 & 0  & 1 & 2  \\
2 & -1 & 0 & 1  \\ 
1  & -2 & -1 & 0 \end{array} \right|$ = 16
$\left|  \begin{array}{cccc}
0  & 1  & 2 & -1 \\
-1 & 0  & 1 & -2  \\
-2 & -1 & 0 & 1  \\ 
1  & 2 & -1 & 0 \end{array} \right|$ = 16
$\left|  \begin{array}{cccc}
0  & 1  & -2 & -1 \\
-1 & 0  & 1 & -2  \\
2 & -1 & 0 & 1  \\ 
1  & 2 & -1 & 0 \end{array} \right|$ = 16
but,
$\left|  \begin{array}{cccc}
0  & 1  & 2 & 1 \\
-1 & 0  & 1 & -2  \\
-2 & -1 & 0 & 1  \\ 
-1  & 2 & -1 & 0 \end{array} \right|$ = 36.
In the above example the value of determinant doesn't changed when the elements 2 and -2 are interchanged but the value of determinant changed when the elements 1 and -1 are interchanged.
Kindly any one answer me when does the value of determinant change?

I suspect the answer to "when does the value of the determinant change?" is "nearly always".  Consider what happens when you interchange the $(1,3)$ and $(3,1)$ elements.  Doing the algebra (by computer or by hand) gives
$$\eqalign{\det\pmatrix{0&a&b&c\cr -a&0&d&e\cr-b&-d&0&f\cr -c&-e&-f&0\cr}
  -&\det\pmatrix{0&-a&b&c\cr a&0&d&e\cr-b&-d&0&f\cr -c&-e&-f&0\cr}\cr
  &=-4be(af+cd)\ .\cr}$$
In your example it happens that $af+cd=0$, so the two determinants are the same.  But this will not usually be the case.
