Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Product of a matrix and its adjugate

Builds on Adjugate matrixDeterminant

Statement

Let KK be a commutative ring, let n≥2n \ge 2, and let AA be an n×nn \times n matrix over KK. Then Aadj⁡(A)=adj⁡(A)A=det⁡(A) In,A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n, where adj⁡(A)\operatorname{adj}(A) is the adjugate of AA and det⁡\det is the determinant.

Remarks

The diagonal entries of Aadj⁡(A)A \operatorname{adj}(A) are the Laplace expansions of det⁡A\det A along the rows. The entries off the diagonal are expansions of matrices with two equal rows, so they are zero. The identity holds for all square matrices, also for singular ones.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMinor, definition, no evidence levelMinordefinitionCofactor, definition, no evidence levelCofactordefinitionAdjugate matrix, definition, no evidence levelAdjugate matrixdefinitionProduct of a matrix and its adjugate, statement, no evidence levelProduct of a matrix a…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Linear map definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Minor definition
  11. Cofactor definition
  12. Adjugate matrix definition
  13. Product of a matrix and its adjugate statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Adjugate_matrix, Definition
About this page
Tag0528HKWZ1Z
URIm2://0528HKWZ1Z
Kindstatement, theorem
WritingW0 stub
MSC15A09, 15A15
StatusThis copy of the site has no build data. Run m2 status 0528HKWZ1Z.
UsesAdjugate matrix, Determinant

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