---
kind: statement
tag: "16JNSD7YBW"
name: The effect of row operations on the determinant
role: proposition
msc:
  - 15A15
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-4.xml#L846
    locator: src/section3-4.xml:846
uses:
  - 0C3CFHDN6R
  - 04XWH28452
about:
  - 00KDZRW4C0
---

## Statement

Let $A$ be a square matrix, and let $A'$ be a matrix obtained from $A$ by one elementary [[row operation|04XWH28452]]. Write $\det$ for the [[determinant|00KDZRW4C0]].

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

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

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

## Remarks

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