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6.9: Untitled Page 132

  • Page ID
    18265
  • Chapter 6

    III. Any two rows in the atomic matrix can be

    interchanged without affecting the system of

    equations.

    The column / ow

    r

    interchange operation that we will use in the treatment of Eq. 6‐22 is described as follows:

    IV. Any two columns in the atomic matrix can be

    interchanged without affecting the system of

    equations provided that the corresponding rows of the column matrix of net rates of production are also

    interchanged.

    In terms of Eq. 6‐20 this latter operation can be described mathematically as N R

    N R ,

    B, D  1 , 2 ,...,N ,

    J  1 , 2 ,...,T

    (6‐40)

    JB B

    JD D

    We can use these operations to develop row equivalent matrices, row reduced matrices, row echelon matrices, and row reduced echelon matrices. In order to illustrate these concepts, we consider the following example of Axiom II:

    R

    1

    R

    2

    2 2 0 2 0 4  

    6 4 2 4 2 6

    R

     3 

    Axiom II:

     0

    (6‐41)

    2 2 0 1 1 3 R

     4 

    1 0 1 1 0 0 

    R

    5

     R

    6 

    Directing our attention to the atomic matrix, we subtract three times the first row from the second row to obtain a row equivalent matrix given by

    R

    1

    R

    2 

     2

    2

    0

    2

    0

    4  

    0 2

    2 2

    2 6

    R

     3 

    R 2  3 1

    R :

     0

    (6‐42)

    2

    2

    0

    1

    1

    3  R

     4 

     1

    0

    1

    1

    0

    0 

      R 5

    R

     6 

    Stoichiometry

    243

    Dividing the first row by two will create a coefficient of one in the first row of the first column. This operation leads to

    R

    1

    R

    2 

     1

    1

    0

    1

    0

    2  

    0 2

    2 2

    2 6

    R

     3 

    1

    R 2 :

     0

    (6‐43)

     2

    2

    0

    1

    1

    3  R

     4 

    1

    0

    1

    1

    0

    0  R

    5

    R

     6 

    Multiplication of the second row by 1 2 provides

    R

    1

    R

    2 

     1

    1

    0

    1

    0

    2  

    0

    1 1

    1

    1

    3

    R

     3 

    R 2 1 2 :

     0

    (6‐44)

     2

    2

    0

    1

    1

    3  R

     4 

    1

    0

    1

    1

    0

    0  R

    5

    R

     6 

    Using several elementary row operations, we construct a row echelon form of the atomic matrix that is given by

    R

    1

    R

    2

    1

    1

    0

    1

    0

    2  

    0

    1 1

    1 1

    3

    R

     3 

     0

    (6‐45)

     0

    0

    0

    1 1

    1  R

     4 

    0

    0

    0

    0

    0

    0  R

    5

    R

     6 

    The row of zeros indicates that one of the four equations represented by Eq. 6‐41

    is not independent, i.e., it is a linear combination of two or more of the other equations. This means that the rank of the atomic matrix represented in Eq. 6‐41

    is three, r  rank N   3

    JA

    .

    We can make further progress toward the row reduced echelon form by subtracting row #2 from row #1 to obtain

    244