Elementary Row Operations

Elementary Row Operations

Elementary row operations:

Replace: replace one row by the sum of that row and a constant multiple of another row.

Swap: interchange two rows

Scale: multiply a row by a non-zero constant

Let's use the elementary row operations to reduce a matrix to row echelon form.

In[1]:=

In[2]:=

Out[3]//MatrixForm=

In[4]:=

Out[5]//MatrixForm=

In[6]:=

Out[7]//MatrixForm=

Elementary Matrices

Define procedures which create elementary matrices.

Premultiplication of a given matrix by an elementary matrix effects an elementary row operation.

In[8]:=

Testing. Create some elementary matrices.

In[11]:=

Out[12]//MatrixForm=

Out[13]//MatrixForm=

Out[14]//MatrixForm=

Echelon Form

Use elementary row operations to reduce a matrix to row echelon form.

Premultiplication by elementary matrices accomplishes the transformation.

In[15]:=

Out[19]//MatrixForm=

In[20]:=

Out[22]//MatrixForm=

Reduced Echelon Form

Use row reduction to solve a system of linear equations.

In[23]:=

Out[25]//MatrixForm=

Let's check that.

In[26]:=

In[27]:=

Out[28]//MatrixForm=

Out[30]//MatrixForm=

Out[32]//MatrixForm=

Out[33]=

Created by Mathematica (December 15, 2004) |