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.