I have seen in a lot of Texts and Websites where the determinant for a square matrix is gotten by obtaining minors along the first row which seems to be the norm. but recently I see it can be gotten by obtaining minors along the last column or across the diagonal(I see this a lot for obtaining the determinants for 4x4 matrices). so I decided to try this for the following 3x3 matrix(A)
$$A = \begin{pmatrix} 1^+&2^-&1^+\\ 3^-&-4^+&-2^-\\ 5^+&3^-&5^+\\ \end{pmatrix}$$
$(i)$ taking minors across the first row of $A$(the usual norm)
$$\det(A) = \begin{array}{c|cc|} 1&-4&-2\\ & 3&5\\ \end{array} + \begin{array}{c|cc|} -2&3&-2\\ & 5&5\\ \end{array} + \begin{array}{c|cc|} 1&3&-4\\ & 5&3\\ \end{array} = -35 $$
$(ii)$ taking minors across the last column of $A$ $$\det(A) = \begin{array}{c|cc|} 1&3&-4\\ & 5&3\\ \end{array} + \begin{array}{c|cc|} -(-2)&1&2\\ & 5&3\\ \end{array} + \begin{array}{c|cc|} 5&1&2\\ & 3&-4\\ \end{array} = -35 $$
$(iii)$ taking minors across the diagonal of $A$
$$\det(A) = \begin{array}{c|cc|} 1&-4&-2\\ & 3&5\\ \end{array} + \begin{array}{c|cc|} -(-4)&1&1\\ & 5&5\\ \end{array} + \begin{array}{c|cc|} 5&1&2\\ & 3&-4\\ \end{array} = -64 $$
Why do I get a different value across the diagonal if the determinant for any given matrix is constant irrespective of where the minors are obtained or are they all correct?
P.S: case $(ii)$ result was corrected
So to answer your question: you get different values for two different reasons. You are right that the determinant value of a matrix is independent of which minor we choose. Note however that choosing the diagonal is not a valid method for this. There are ways to compute the determinant using the diagonal, but not in the way you use here.
Secondly, your computation of (ii) is wrong, as pointed out in the comments. The reason the first row is usually chosen is for ease of methodology. If however you are working with a mtarix with zeros as entries it can be beneficial to choose that row or column to work with.