# 6.9: Untitled Page 132

- Page ID
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## 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

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