Math 2.0

Linear and multilinear algebra; matrix theory Corollary

Inverse in terms of the adjugate

Builds on Product of a matrix and its adjugateInvertible matrix

Statement

Let KK be a commutative ring, let n≥2n \ge 2, and let AA be an n×nn \times n matrix over KK such that det⁡A\det A is a unit of KK. Then AA is invertible, and A−1=(det⁡A)−1adj⁡(A),A^{-1} = (\det A)^{-1} \operatorname{adj}(A), where adj⁡(A)\operatorname{adj}(A) is the adjugate of AA.

Remarks

The statement follows by dividing Aadj⁡(A)=adj⁡(A)A=det⁡(A)InA \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A) I_n by det⁡A\det A. For n=2n = 2 and ad−bc≠0ad - bc \ne 0 over a field, (abcd)−1=1ad−bc(d−b−ca)\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. For large nn, row reduction computes the inverse much faster than this formula.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionMultilinear 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…statementInverse in terms of the adjugate, statement, no evidence levelInverse in terms of t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Matrix multiplication definition
  5. Square matrix definition
  6. Transpose definition
  7. Linear map definition
  8. Invertible matrix definition
  9. Multilinear map definition
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Minor definition
  13. Cofactor definition
  14. Adjugate matrix definition
  15. Product of a matrix and its adjugate statement
  16. Inverse in terms of the adjugate statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Adjugate_matrix, Important consequence
About this page
Tag0ENVM493VK
URIm2://0ENVM493VK
Kindstatement, corollary
WritingW0 stub
MSC15A09
StatusThis copy of the site has no build data. Run m2 status 0ENVM493VK.
UsesProduct of a matrix and its adjugate, Invertible matrix

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.