1) Switching two rows or columns causes the determinant to switch sign
2) Adding a multiple of one row to another causes the determinant to remain the same
3) Multiplying a row as a constant results in the determinant scaling by that constant.
Using the geometric definition of the determinant as the area spanned by the columns of the matrix, could someone give me a geometric interpretation of the above theorems? Thanks.
Best Answer
Row operations can be thought of as acting on the ENTIRE space by reflection, shearing, or dilation.
For 2, have you heard of Cavalieri's principle? It says that shearing things while holding each cross section steady maintains volume.
For 3, you are just stretching or shrinking along each axis.
For 1, you are just reflecting in the plane $x_i=x_j$.
Edit: Every row operation is the effect of multiplying on the left by an elementary matrix. We can think of this elementary matrix as a map from $R^n$ to $R^n$, which changes every vector in $R^n$ including the column vectors. Thus, each row operation corresponds to a way of changing the whole space.
The row operation in 1 interchanges two rows. This corresponds to interchanging two coordinates in the space. It is not obvious, but it has been shown that interchanging two coordinates is the same thing as reflecting the entire space around the subset where the two coordinates are equal. This does not change volume.
The row operation in 3 corresponds to stretching one coordinate by the multiple given, which multiplies volume by the same amount.
The operation in 2 can be thought of as follows: say that you add a multiple of the second row to the first. Imagine the space sliced into 'pancakes', one for each value of the second coordinate. The map doesn't interchange pancakes, it just slides each pancake 'horizontally'. This doesn't change the volume of anything.