1_elementary_row_operations.html

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=

( {{1, -2, 1, 0}, {0, 4, -10, 8}, {0, 0, 11/2, -3}} )

Elementary Matrices

Define procedures which create elementary matrices.
Premultiplication of a given matrix by an elementary matrix effects an elementary row operation.

In[8]:=

$eReplace[n_, i_, a_, j_] := Module[{e = IdentityMatrix[n]}, e[[i]] = e[[i]] + a e[[j]] ; e] $

$eSwap[n_, i_, j_] := Module[{e = IdentityMatrix[n], temp},  temp = e[[i]] ; e[[i]] = e[[j]] ; e[[j]] = temp ; e] $

$eScale[n_, i_, a_] := Module[{e = IdentityMatrix[n]}, e[[i]] = a e[[i]] ; e]$

Testing. Create some elementary matrices.

In[11]:=

a = ({{1, -2, 1, 0}, {2, 0, -8, 8}, {-4, 5, 9, -9}}) ; 

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]:=

a = ({{1, -2, 1, 0}, {2, 0, -8, 8}, {-4, 5, 9, -9}}) ; 

Out[19]//MatrixForm=

In[20]:=

Out[22]//MatrixForm=

( {{1, -2, 1, 0}, {0, 4, -10, 8}, {0, 0, 11/2, -3}} )

Reduced Echelon Form

Use row reduction to solve a system of linear equations.

In[23]:=

a = ({{1, -2, 1, 0}, {0, 2, -8, 8}, {-4, 5, 9, -9}}) ; 

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)