Lemma 10.15.6. Let R be a ring. Let n \geq m. Let A = (a_{ij}) be an n \times m matrix with coefficients in R, written in block form as
A = \left( \begin{matrix} A_1
\\ A_2
\end{matrix} \right)
where A_1 has size m \times m. Let B be the adjugate (transpose of cofactor) matrix to A_1. Then
AB = \left( \begin{matrix} f 1_{m \times m}
\\ C
\end{matrix} \right)
where f = \det (A_1) and c_{ij} is (up to sign) the determinant of the m \times m minor of A corresponding to the rows 1, \ldots , \hat j, \ldots , m, i.
Proof.
Since the adjugate has the property A_1B = B A_1 = f the first block of the expression for AB is correct. Note that
c_{ij} = \sum \nolimits _ k a_{ik}b_{kj} = \sum (-1)^{j + k}a_{ik} \det (A_1^{jk})
where A_1^{ij} means A_1 with the jth row and kth column removed. This last expression is the row expansion of the determinant of the matrix in the statement of the lemma.
\square
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