Let be a monic polynomial (i.e. ) and let where if , 1 if , and 0 otherwise. Show that .
Recall that , where if and 0 otherwise.
Using the Cofactor Expansion Formula along the th column (derived here), we have , where denotes the matrix minor of .
Evidently, we have that if , if , if and , if and , and 0 otherwise. So , where is lower triangular of dimension and diagonal entries equal to and is upper triangular of dimension and diagonal entries equal to . So we have .
So we have as desired.