Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant of the inverse

Builds on DeterminantInvertible matrixDeterminant is multiplicative

Statement

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

Remarks

The proof applies the product rule to AA−1=InA A^{-1} = I_n: det⁡A⋅det⁡(A−1)=1\det A \cdot \det(A^{-1}) = 1. Over a field, this says det⁡(A−1)=1/det⁡A\det(A^{-1}) = 1 / \det A.

Reading path

To understand this page from the ground up, read these 12 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 matrixdefinitionLinear 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 levelDeterminantdefinitionDeterminant is multiplicative, statement, no evidence levelDeterminant is multip…statementDeterminant of the inverse, statement, no evidence levelDeterminant of the in…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. Linear map definition
  7. Invertible matrix definition
  8. Multilinear map definition
  9. Alternating multilinear map definition
  10. Determinant definition
  11. Determinant is multiplicative statement
  12. Determinant of the inverse statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Invertible_matrix, Other properties
About this page
Tag0783S9D0ZH
URIm2://0783S9D0ZH
Kindstatement, proposition
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 0783S9D0ZH.
UsesDeterminant, Invertible matrix, Determinant is multiplicative

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.