How To Find TheHow To Find TheReduced RowReduced RowEchelon FormEchelon Form

Reduced Row Reduced Row Echelon FormEchelon Form

A matrix is said to be in A matrix is said to be in reduced row echelon reduced row echelon

form provided it form provided it satisfies the following satisfies the following

properties:properties:

All zero rows, if All zero rows, if there are any, there are any,

appear as bottom appear as bottom rows.rows.

The first nonzero The first nonzero entry in a nonzero entry in a nonzero

row is a 1; it is row is a 1; it is called a leading 1.called a leading 1.

For each nonzero For each nonzero row, the leading 1 row, the leading 1

appears to the appears to the right and below right and below

any leading 1s in any leading 1s in preceding rows.preceding rows.

If a column If a column contains a leading contains a leading 1, then all other 1, then all other entries in that entries in that

column are zero.column are zero.

There are three There are three elementary row elementary row

operations allowed:operations allowed:• Interchanging rows of a matrix;Interchanging rows of a matrix;

•Multiply one row by a number Multiply one row by a number that is not equal to 0;that is not equal to 0;

•Add any number not equal to 0) Add any number not equal to 0) times one row of the matrix to times one row of the matrix to another row (as long as the two another row (as long as the two rows are not equal to each other).rows are not equal to each other).

Allow the following to equal Allow the following to equal A:A:

174

402

351

Allow the following to equal b, Allow the following to equal b, the solutions to each row of A:the solutions to each row of A:

2

6

7

The augmented matrix (Ab) is The augmented matrix (Ab) is below and this is the matrix we below and this is the matrix we will be reducing to reduced row will be reducing to reduced row

echelon form:echelon form:

2174

6402

7351

Below is the original matrixBelow is the original matrix

2174

6402

7351

Add -2 times the first row to the Add -2 times the first row to the second rowsecond row

-2(R1)+R2=new R2-2(R1)+R2=new R2

2174

2010100

7351

Add -4 times the first row to the Add -4 times the first row to the third rowthird row

-4(R1)+R4=new R4-4(R1)+R4=new R4

2613130

2010100

7351

Multiply the second row by -1/10Multiply the second row by -1/10 (-1/10)R2=new R2(-1/10)R2=new R2

2613130

2110

7351

Add 13 times the second row to the Add 13 times the second row to the third rowthird row

(13)R2+R3=new R3(13)R2+R3=new R3

0000

2110

7351

Add -5 times the second row to the Add -5 times the second row to the first rowfirst row

(-5)R2+R1=new R1(-5)R2+R1=new R1

The bottom row is comprised of all 0’s which means that when we are solving the system of linear equations, that

row means nothing.

0000

2110

3201

To solve, the system of equations, To solve, the system of equations, set each row in A equal to the set each row in A equal to the

corresponding row in b and add in corresponding row in b and add in the variables of x (x1, x2 and x3):the variables of x (x1, x2 and x3):

0000

2110

3201

321

321

321

xxx

xxx

xxx

This then is simplified to:This then is simplified to:

00

211

321

32

31

xx

xx

0=00=0

No equation of this system No equation of this system has a form has a form zero = zero = nonzerononzero; Therefore, the ; Therefore, the system is system is consistentconsistent (has (has at least one solution).at least one solution).

Solving for the linear Solving for the linear equations:equations:

x1=-2 x3-3

x2=-1 x3+2

x3=arbitrary

Solution:Solution:

The system has The system has infinitely many infinitely many solutionssolutions because each of the because each of the variables is equal to the sum of a variables is equal to the sum of a constant and a variable, therefore constant and a variable, therefore making it impossible for one solution making it impossible for one solution to be correct.to be correct.