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Matrix/Adjugate matrix/Formula with determinant/Fact/Proof

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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.