Math 2.0

Linear and multilinear algebra; matrix theory Determinants Theorem

Determinant is multiplicative

Builds on DeterminantMatrix multiplication

Statement

Let KK be a commutative ring and let AA and BB be n×nn \times n matrices over KK. Then det⁡(AB)=det⁡A⋅det⁡B,\det(AB) = \det A \cdot \det B, where ABAB is the matrix product and det⁡\det is the determinant.

Remarks

Thus det⁡\det is a group homomorphism from the general linear group GLn(F)GL_n(F) to F×F^\times, over a field FF. It follows that det⁡(AB)=det⁡(BA)\det(AB) = \det(BA), although AB≠BAAB \ne BA in general. Similar matrices have the same determinant, because det⁡(P−1AP)=det⁡A\det(P^{-1} A P) = \det A.

Reading path

To understand this page from the ground up, read these 10 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 mapdefinitionMultilinear 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…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. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Determinant is multiplicative statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules
About this page
Tag04CPJ6AF00
URIm2://04CPJ6AF00
Kindstatement, theorem
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 04CPJ6AF00.
UsesDeterminant, Matrix multiplication

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

Used by