Linear equations MOC

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

  1. Swapping of rows (changing the order of equations): π‘Ÿ1 β†”π‘Ÿ2
  2. Scalar multiplication of rows: π‘Ÿ1 ←2π‘Ÿ1
  3. Addition of rows: π‘Ÿ1 ←3π‘Ÿ1 +2π‘Ÿ2

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.

  1. Swapping of rows (π‘Ÿ1 β†”π‘Ÿ2) gives
det(𝐴′)=βˆ’det(𝐴)
  1. Scalar multiplication (π‘Ÿ1 β†π›Όπ‘Ÿ1) gives
det(𝐴′)=𝛼det(𝐴)
  1. Addition of rows (π‘Ÿ1 β†π›Όπ‘Ÿ1 +π›½π‘Ÿ2) gives
det(𝐴′)=𝛼det(𝐴)

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.

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