# 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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