Let be a square matrix over a field . Formulate and prove the cofactor expansion formula for along the th column.
We begin with a definition. If , where has dimension , then the -minor of is the matrix .
Recall that the cofactor expansion formula for along the th row is
The analogous expansion along the th column is
which we presently prove to be true. First, note that ; this follows from our definition of minors and the fact that . Now we have the following.