# adjugate

The *adjugate ^{}*, $\mathrm{adj}(A)$, of an $n\times n$
matrix $A$, is the $n\times n$ matrix

$$\mathrm{adj}{(A)}_{ij}={(-1)}^{i+j}{M}_{ji}(A)$$ | (1) |

where ${M}_{ji}(A)$ is the indicated minor of $A$ (the determinant^{}
obtained by deleting row $j$ and column $i$ from $A$). The adjugate
is also known as the *classical adjoint*, to distinguish it from
the usual usage of “adjoint^{}” (http://planetmath.org/AdjointEndomorphism) which
denotes the conjugate transpose^{} operation^{}.

An equivalent^{} characterization of the adjugate is the following:

$$\mathrm{adj}(A)A=det(A)I.$$ | (2) |

The equivalence of (1) and (2) follows easily
from the multi-linearity
properties (http://planetmath.org/DeterminantAsAMultilinearMapping) of the determinant.
Thus, the adjugate operation is closely related to the matrix inverse.
Indeed, if $A$ is invertible^{}, the adjugate can be defined as

$$\mathrm{adj}(A)=det(A){A}^{-1}$$ |

Yet another definition of the adjugate is the following:

$\mathrm{adj}(A)={p}_{n-1}(A)I$ | $-{p}_{n-2}(A)A+{p}_{n-3}(A){A}^{2}-\mathrm{\cdots}$ | (3) | ||

$+{(-1)}^{n-2}{p}_{1}(A){A}^{n-2}+{(-1)}^{n-1}{A}^{n-1},$ |

where ${p}_{1}(A)=\mathrm{tr}(A),{p}_{2}(A),\mathrm{\dots},{p}_{n}(A)=det(A)$
are the elementary invariant polynomials of
$A$. The latter arise as
coefficients in the
characteristic polynomial^{} $p(t)$ of $A$, namely

$$p(t)=det(tI-A)={t}^{n}-{p}_{1}(A){t}^{n-1}+\mathrm{\cdots}+{(-1)}^{n}{p}_{n}(A).$$ |

The equivalence of (2) and (3) follows from
the Cayley-Hamilton theorem^{}. The latter states that $p(A)=0$, which
in turn implies that

$$A({A}^{n-1}-{p}_{1}(A){A}^{n-2}+\mathrm{\cdots}+{(-1)}^{n-1}{p}_{n-1}(A))={(-1)}^{n-1}det(A)I$$ |

The adjugate operation enjoys a number of notable properties:

$\mathrm{adj}(AB)=\mathrm{adj}(B)\mathrm{adj}(A),$ | (4) | ||

$\mathrm{adj}({A}^{t})=\mathrm{adj}{(A)}^{t},$ | (5) | ||

$det(\mathrm{adj}(A))=det{(A)}^{n-1}.$ | (6) |

Title | adjugate |
---|---|

Canonical name | Adjugate |

Date of creation | 2013-03-22 13:09:42 |

Last modified on | 2013-03-22 13:09:42 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 17 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A09 |

Synonym | classical adjoint |