I have been learning about determinants and, in particular, calculating determinants by adding multiples of one row/column to another. However on doing some problems, I see that you cannot simply add and subtract these anyhow.
you cannot transform row $i$ by subtracting some multiple of row $j$, and then transform row $j$ by subtracting some multiple of row $i$ (as in, what $i$ was before you transformed it). The resulting matrix will have a different determinant to help original.
secondly, you cannot add/subtract multiples of one row from itself without changing the determinant.
I know these points are probably very obvious and elementary, however they were not at all obvious to me from how I was taught (i.e. that you can just subtract multiples of rows/columns to obtain the triangular form.)
Although I cannot come up with a geometric/intuitive picture of why the above statements are true, I can motivate it in the $3\times 3$ case considering the determinant as the vector triple product:
$det (A)=a \cdot (b \times c) $
I can see from this that if I add a multiple of a vector to itself, the result is changed. I can only add multiples of different rows. Also, I can see that the result is also changed if I add a multiple of one row to another, and simultaneously add a multiple of the second row to the first (say change $b$ to $b+2a$, and $a$ to $a+0.5b$ etc)
However I do not know how to rationalise this for higher dimension matrices. Is there an equivalent cross /wedge product formulation for the determinant in higher dimensions that would illustrate this easily? The only formulation if the determinant I do know is involving the cofactor, or the levi civita tensor and cycling around the elements. But it is difficult to see any result in these forms, as you can with the cross product formula.