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salutian?

7.2 ELEMENTARY ROW OPERATIONS

The focus of our concern in the last section was on the coefficients a, and hi of

fbc systems with which WC worked. In fact, it was a little inefficient to rewrite

the x,â€˜s, the plus signs, and the equal signs each time we transformed a system.

It makes sense to simplify the representation of linear system (2) by writing two

rectangular arrays of its cocflicients, called matrices. The first array is

which is called the add on a column corre-

coefficient matrix of (2). When we

sponding t o the right-hand side in system (2), we obtain the matrix

which is called the of (2). The mws oft? correspond naturally

augmented matrix

t o the equations of (2). For example,

1

( 1

-2

and (: -: i)

3 1

arc the coefficient matrix and the augmented matrix of system (7). For accounting

purposes, it is often helpful to drew a vertical line just before the last column of

the augmented matrix, where the = signs would naturally appear, e.g.,

-2 I 8

1 I 31

(:

Our three clcmentary equation operations now become elementary row opera-

tions:

(I) interchange t w o rows of il matrix.

(2) change a row by adding t o it a multiple of another row. and

(3) multiply each element in a how by the same M˜LCIO numhcr,

The new augmrntcd matrix will reprcscnt a system of linear equations which is

equivalent to the system rcprcsented by the old augmented matrix.

To see this equivalence, first ohserve that each clemrntary row operation can

hc reversed. Clearly the interchanging of two rows or the multiplication of a row

hy a nonzero scalar can be reversed. Suppose we consider the operation in

TOW

k times the second row of the augmented matrix A is added to the first row

which

of/i The new augmented matrix is

B=

â€˜.

ox,, ,..

-k times the second row t o the first t o w Y

However. if start with R and add

WC

w t : Cll recover;\. Thus the TOM operation can be reversed. Since elementary row

operations correspond to the three operations of adding a multiple of one equation

to another equation, multiplying both sides of an equation by the same scalar,

and changing the order of the equations, any solution to the original system of

equations will be a solution to the transformed system. Since these operations

are reversible, any solution to the transformed system of equations will also be a

solution to the original system. Consequently the systems represented by matrices

a and B have idenrical solution sets; they are equivalent.

The goal of performing row operations is to end up with a matrix that looks

much like (10). The nice feature about the augmented matrix representing (10)

-0.4 -0.3 I 130

1

0 0.8 -0.2 I 100 (1%

0.7 1 210 1

(0 0

hegim with more zeros than does the previous

is that each row Such a matrix

TOW.

is said to be in row echelon form.

A row of a matrix is said to have kleadingzeros if the first k elements

Definition

of the row are all zeros and the (k + 1)th element of the row is not zero. With this

terminology, a matrix is in row echelon form if each row has more leading zeros

than the row preceding it.

The first TOW of the augmented matrix (13) has no leading zeros. The second

row has one. and the third row has two. Since each row has more leading zeros

than the previous row, matrix (13) is in TOW echelon form. Letâ€™s look at some more

concrete examples.

Ewnple 7.2 The matrices

sre in TOW echelon form. If a matrix in row echelon form has a row containing

only zero\; thrn all the subsequent rows must contain only ZCKX

Eruml˜le 7..? The matrices

132 SYSTEMS OF LINEAR EQUATIONS [71

Example 7.4 The matrix whose diagonal elements (a;iâ€™s) are Is and whose off-

diagonal elements (a,â€˜˜ with i not equal to j) are all OS is in row echelon form.

This matrix arises frequently throughout linear algebra, and is called the identity

matrix when the number of rows is the same as the number of columns:

Example 7.5 The matrix each of whose elements is 0 is called the zero matrix

Iand is in row echelon form:

The usefulness of row echelon form can he seen by considering the system of

equations (8). The augmented matrix associated with (8) is

-0.4

1

-0.2 0.88 -0.14 I

( PO.5 -0.2

and through various row operations we reduced it to

1 ˜0.4 -0.3 I 130

-0.2 I 1 0 0 (14)

0 0.8

0 0 0.7 I 210 1

This last matrix is in row echelon form and the corresponding system can he easily

solved by substitution. Simply rewrite it in equation form and solve it from hottom

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