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*Definition*: Let \(A=[a_{ij}]\) be an \(n \times n\) matrix over a field \(K\) and let \(A_{ij}\) denote the cofactor of \(a_{ij}\). The classical adjoint of \(A\), denoted by adj \(A\), is the transpose of the matrix of cofactors of \(A\). Namely,

We say "classical adjoint" instead of simple "adjoint" because the term "adjoint" is currently used for an entirely different concept.

In order to understand what is a cofactor, consider an n-square matrix \(A=[a_{ij}]\). Let \(M_{ij}\) denote the \((n-1\))-square submatrix of \(A\) obtained by deleting its \(i\)th row and \(j\)th column. The determinant \(|M_{ij}|\) is called the minor of the element \(a_{ij}\) of \(A\), and we define the cofactor of \(a_{ij}\), denoted by \(A_{ij}\), to be the "signed" minor: \[A_{ij}=(-1)^{i+j}|M_{ij}|\].

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