Proof
Let
.
Let the coefficients of the adjugate matrix be denoted by
-
The coefficients of the product are
-
In case
,
this is , as this sum is the expansion of the determinant with respect to the -th column. So let
,
and let denote the matrix that arises from by replacing in the -th column by the -th column. If we expand with respect to the -th column, then we get
-
Therefore, these coefficients are , and the first equation holds.
The second equation is proved similarly, where we use now the expansion of the determinant with respect to the rows.