So normally the formula to calculate the inverse of a 3x3 matrix is to transpose the matrix and calculate its minors' determinants then switch the sign for every second element and multiply each element with 1/det of the original matrix. As seen on the image attached, Wolfram uses another approach, that totally works,but I can't really figure out the rules in it. It yields the determinants correctly therefore no sign switches are required, so I figure it combines the two steps somehow? But how to come up with this formula? Thanks in advance
Look at the second entry of the first row of that matrix. In the classical Cramer's Rule approach you describe, that would be the cofactor for the row-2/column-1 entry of $A$, i.e., the determinant of the matrix gotten by deleting the 2nd row and first column, i.e. $$ \det \pmatrix{a_{12} & a_{13} \\ a_{32} & a_{33}} $$ which would then be multiplied by $-1$. But in the formula above, the two columns of the matrix are swapped, which achieves the multiplication by $-1$.