## Column operations

Let be an square matrix composed of column vectors  . Then we have the following properties for determinants.

1. (for ).

2. (where is another column vector operated on the th column of ).

Properties 3 and 4 can be immediately verified by Laplace expansions. When we apply Laplace expansions recursively for , we end up with the determinants of matrices of the respective forms

and

If , we have , which shows Property 2. If , we have , which shows Property 5. Finally Properties 4 and 5 together implies Property 1.

Department of Mathematics
Last modified: 2005-10-21