Editing Adjugate matrix (section)

{{Short description|For a square matrix, the transpose of the cofactor matrix}}
In [[linear algebra]], the '''adjugate''' or '''classical adjoint''' of a [[square matrix]] {{math|'''A'''}}, {{math|adj('''A''')}}, is the [[transpose]] of its [[cofactor matrix]].<ref>{{cite book |first=F. R. |last=Gantmacher |author-link=Felix Gantmacher |title=The Theory of Matrices |volume=1 |publisher=Chelsea |location=New York |year=1960 |isbn=0-8218-1376-5 |pages=76–89 |url=https://books.google.com/books?id=ePFtMw9v92sC&pg=PA76 }}</ref><ref>{{cite book |last=Strang |first=Gilbert |title=Linear Algebra and its Applications |publisher=Harcourt Brace Jovanovich |year=1988 |isbn=0-15-551005-3 |edition=3rd |pages=[https://archive.org/details/linearalgebraits00stra/page/231 231–232] |chapter=Section 4.4: Applications of determinants |author-link=Gilbert Strang |chapter-url=https://archive.org/details/linearalgebraits00stra/page/231 |chapter-url-access=registration}}</ref> It is occasionally known as '''adjunct matrix''',<ref>{{cite journal|author1=Claeyssen, J.C.R.|year=1990|title=On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions|journal=Journal of Sound and Vibration|volume=140|issue=1|pages=73–84|doi=10.1016/0022-460X(90)90907-H|bibcode=1990JSV...140...73C }}</ref><ref>{{cite journal|author1=Chen, W.|author2=Chen, W.|author3=Chen, Y.J.|year=2004|title=A characteristic matrix approach for analyzing resonant ring lattice devices|journal=IEEE Photonics Technology Letters|volume=16|issue=2|pages=458–460|doi=10.1109/LPT.2003.823104|bibcode=2004IPTL...16..458C }}</ref> or "adjoint",<ref>{{cite book|first=Alston S.|last=Householder|title=The Theory of Matrices in Numerical Analysis |publisher=Dover Books on Mathematics|year=2006|author-link=Alston Scott Householder | isbn=0-486-44972-6 |pages=166–168 }}</ref> though that normally refers to a different concept, the [[Hermitian adjoint|adjoint operator]] which for a matrix is the [[conjugate transpose]].

The product of a matrix with its adjugate gives a [[diagonal matrix]] (entries not on the main diagonal are zero) whose diagonal entries are the [[determinant]] of the original matrix:
:<math>\mathbf{A} \operatorname{adj}(\mathbf{A}) = \det(\mathbf{A}) \mathbf{I},</math>
where {{math|'''I'''}} is the [[identity matrix]] of the same size as {{math|'''A'''}}. Consequently, the multiplicative inverse of an [[invertible matrix]] can be found by dividing its adjugate by its determinant.