Elementary row operation
An elementary row operation is any one of three operations which may be performed on a system of linear equations without changing the solution: linalg
- Swapping of rows (changing the order of equations):
- Scalar multiplication of rows:
- Addition of rows:
These operations are used by both Gaußian elimination, and the more advanced Gauß-Jordan elimination.
In the sense that these operations may be performed on a system without altering it, a homogenous system may therefore be seen as an example of a vector space, with the addition of equations and their multiplication by a scalar as its two operations.
Effects on determinant
Each elementary row operation affects the Matrix determinant in predictable ways.
- Swapping of rows (
) gives
\begin{align*} \det(A’) = -\det(A) \end{align*}
\begin{align*} \det(A’) = \alpha \det(A) \end{align*}
This is useful, since a upper-triangle matrix (i.e. Row echelon form) has a determinant equal to the product of the main diagonal. Hence Gaußian elimination provides a useful method for finding the Matrix determinant. These properties can be derived by considering the Exterior algebra.