Determinant changes through elementary row operations demonstrations

Question

I just didn't manage to find a demonstration I can understand about the fact.

I would like to have a demonstration on these properties.

1. Exchanging two rows changes determinant sign

2. If I have a Matrix A, then if a row a -> 3a, also does det(A)

3. The other elementary row operation has no efect on det(A)

Thank you very much in advance.

PD: My knowledge of Lineal Algebra is nearly 0, I just began the course.

Answer 1

The determinant is the signed volume of the image of the parallelepiped spanned by any orthonormal basis under the linear transformation. Nevermind any other definition you might have heard, that is the one to understand what is going on.

What do I mean by that? Well, take the set of edges $e_1,e_2,e_3,\ldots$ where $e_i$ is the $i$-th unit vecotr - the vector with only zeros and a 1 at position $i$. They span a parallelepiped (in three dimensions, it's just a cube, in two it's a square and so on).

Now, the $i$-th row of a matrix defines the vector that $e_i$ is mapped to. Hence the parallelepiped defined by the vectors $e_1,e_2,\ldots$ is mapped to the one defined by the vectors $Ae_1,Ae_2,\ldots$. The determinant is nothing but the volume of this parallelepiped. Okay, that's not quite right, because the determinant can be negative. It's the signed volume.

The sign is defined by the order of the vectors and it's a little awkward at first. For three dimensional vector spaces, it's easy to relate: If you try to show $e_1,e_2,e_3$ with your hand denoting the first vector with your thumb, the second with your index and the third with your middle finger (finger tips in the direction of the vectors), then you have to use your right hand. Now do the same for the vecotrs $Ae_1,Ae_2,Ae_3$. If you need to use your right hand, the determinant is positive, if you need to use your left, it's negative.

• This gives you a heuristic for the first point: Try exchanging two rows in this picture (this corresponds to switching two fingers) and you'll see that if before you could use the right hand, now you have to use the left and vice versa. Hence the sign changes.

This is no proof for higher dimensions, but the idea is similar. The signed volume refers to the orientation and that switches if you switch rows.

• This also tells you what happens if you multiply a row by some number: You multiply one side of the parallelepiped by that number and therefore also the volume.

• Finally, this also tells you why adding rows doesn't matter: Since the volume of the parallelepiped is given by the length of one side times height to some other side times height to some other side etc. Now adding one side to another side doesn't change the height between these two sides, because - by definition - the height is perpendicular to the vector of that side. Since no height changes and the length of the first side stays the same, the volume remains the same. Also, you don't change the orientation of vectors.