Math 2.0

Linear and multilinear algebra; matrix theory Proposition

The effect of row operations on the determinant

Builds on DeterminantElementary row operations

Statement

Let AA be a square matrix, and let A′A' be a matrix obtained from AA by one elementary row operation. Write det⁡\det for the determinant.

If the operation multiplies one row by a scalar kk, then det⁡(A′)=kdet⁡(A)\det(A') = k\det(A).

If the operation swaps two rows, then det⁡(A′)=−det⁡(A)\det(A') = -\det(A).

If the operation adds a multiple of one row to another row, then det⁡(A′)=det⁡(A)\det(A') = \det(A).

Remarks

The three cases give a way to compute a determinant by row reduction. One tracks the scalings and the swaps along the way.

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
Elementary row operations, concept, no evidence levelElementary row operat…conceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare 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 levelDeterminantdefinitionThe effect of row operations on the determinant, statement, no evidence levelThe effect of row ope…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. Elementary row operations concept
  2. Identity matrix object
  3. Matrix definition
  4. Vector space definition
  5. Square matrix definition
  6. Linear map definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. The effect of row operations on the determinant statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-4.xml#L846, src/section3-4.xml:846
About this page
Tag16JNSD7YBW
URIm2://16JNSD7YBW
Kindstatement, proposition
WritingW0 stub
MSC15A15
StatusThis copy of the site has no build data. Run m2 status 16JNSD7YBW.
AboutDeterminant
UsesDeterminant, Elementary row operations
Changed4 November 2026

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.