Matrix Classical Adjoint (or Adjunct) Calculator

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Basic Information

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,

\[ adj A = [A_{ij}]^T \]

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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