Lemma 4.3: We show that if A is diagonal we can order the elements in any way.

dMat=DiagonalMatrix[{d1,d2,d3}];

Exchange d2 and d3

perm={
{1,0,0},
{0,0,1},
{0,-1,0}
};

perm.dMat.Transpose[perm]//MatrixForm

Det[perm]

Exchange d1 and d2

perm={
{0,1,0},
{-1,0,0},
{0,0,1}
};

perm.dMat.Transpose[perm]//MatrixForm

Det[perm]