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I have the following self-explanatory question.

https://1drv.ms/u/s!AsyHs3E_aioxhipb3wSPSX_heN-t

As seen from the above matrices, I start with the "original" matrix, which is a symmetric matrix, meaning that first row and first column represent the same variable, say X, and 2nd row and 2nd column represent the variable Y, following Z, W, V. I first want to move 2nd row in the "original" matrix to 4th row. This operation is shown in the matrix denoted by r24. After this operation, I want to do the same operation on the same columns, meaning that I want to move 2nd column to 4th column as shown in c24. All of these operations are shown with the colored text. The resulting final matrix, which I aim to create, c24, should be symmetric with respect to the variable names. It means that the final matrix has the ordered variable names as X, Z, W, Y, V in columns and rows. In fact, if the above two operations are done correctly, the order of the variables in rows and columns will remain identical.

I like to do all the operations using a Mathematica function such as f[original, 2, 4] to create the final matrix c24.

Thank you all.

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, idx},

 idx = Range[i, j, Sign[j - i]];

 p = Range[1, Length[A]];

 p[[idx]] = RotateLeft[idx];

 A[[p, p]]

 ]



A = Outer[Plus, Range[5], Range[0, 4]];

A // MatrixForm

$left(
begin{array}{ccccc}
1 & 2 & 3 & 4 & 5 \
2 & 3 & 4 & 5 & 6 \
3 & 4 & 5 & 6 & 7 \
4 & 5 & 6 & 7 & 8 \
5 & 6 & 7 & 8 & 9 \
end{array}
right)$

B = f[A,2,4];

B // MatrixForm

$left(
begin{array}{ccccc}
1 & 3 & 4 & 2 & 5 \
3 & 5 & 6 & 4 & 7 \
4 & 6 & 7 & 5 & 8 \
2 & 4 & 5 & 3 & 6 \
5 & 7 & 8 & 6 & 9 \
end{array}
right)$

to move a row from $i$ to $j$:

rowmove[A_?MatrixQ, i_Integer, j_Integer] := Insert[Delete[A, i], A[[i]], j]

to move a column from $i$ to $j$:

colmove[A_?MatrixQ, i_Integer, j_Integer] := Transpose@rowmove[Transpose[A], i, j]

both at the same time:

move[A_?MatrixQ, i_Integer, j_Integer] := colmove[rowmove[A, i, j], i, j]

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]

So to get the above you would specify

ind = indexlist[5,2,4]

f[A_?MatrixQ, i_Integer, j_Integer] := With[{B = Transpose@Insert[Delete[A, i], A[[i]], j]}, 

  Transpose@Insert[Delete[B, i], B[[i]], j]]

f[A, 2, 4] // MatrixForm