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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?

user241757
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1 Answers1

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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.

David
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