Download Section 2.2

Section 2.2
Systems of Liner Equations:
Unique Solutions
The Gauss-Jordan Elimination Method
Operations
1. Interchange any two equations.
2. Replace an equation by a nonzero
constant multiple of itself.
3. Replace an equation by the sum of
that equation and a constant
multiple of any other equation.
Ex. Solve the system
2x  y  z  3
step
1
x yz  2
2y  2 z  2
3y  z  1
2
x y z  2
 x  3 y  3z  0
x yz 2
y  z 1
3 y  z  1
Row 1 (r1)
Row 2 (r2)
Row 3 (r3)
Replace r2 with [r1 + r2]
Replace r3 with [–2(r1) + r3]
Replace r2 with ½(r2)
x yz  2
3
y  z 1
2 z  4
Replace r3 with [–3(r2) + r3]
x yz 2
4
y  z 1
z  2
x y
5
y
Replace r3 with ½(r3)
4
Replace r1 with [(–1)r3 + r1]
 1
Replace r2 with [r2 + r3]
z  2
3
x
6
y
Replace r1 with [r2 + r1]
 1
z  2
So the solution is (3, –1, –2)
Augmented Matrix
*Notice that the variables in the preceding
example merely keep the coefficients in line.
This can also be accomplished using a matrix.
A matrix is a rectangular array of numbers
System
x y z  2
 x  3 y  3z  0
2x  y  z  3
Augmented matrix
 1 1 1 2 



1
3

3
0


 2 1 1 3 
coefficients
constants
Row Operation Notation:
1.Interchange row i and row j
Ri  R j
2.Replace row j with c times row j
cR j
3.Replace row i with the sum of row i and
c times row j
Ri  cR j
Ex. Last example revisited:
Matrix
System
x y z  2
 x  3 y  3z  0
 1 1 1 2 



1
3

3
0


 2 1 1 3 
2x  y  z  3
x yz  2
2y  2 z  2
3y  z  1
R2  R1
R3  ( 2) R1
1 1 1 2 


0
2

2
2


0 3 1 1
x yz 2
y  z 1
1
R
2 2
3 y  z  1
x yz  2
y  z 1
2 z  4
R3  ( 3) R2
x yz 2
y  z 1
z  2
1
R
2 3
 1 1 1 2 


0 1 1 1 
0 3 1 1
1 1 1 2 


0 1 1 1 
0 0 2 4 
1 1 1 2 


0
1

1
1


0 0 1 2 
x y
y
4
 1
R1  ( 1) R3
R2  R3
z  2
3
x
y
 1
R1  R2
z  2
This is in Row- Reduced
Form
1 1 0 4 


0 1 0 1
0 0 1 2 
1 0 0 3 


0
1
0

1


0 0 1 2 
Row–Reduced Form of a Matrix
1. Each row consisting entirely of zeros lies
below any other row with nonzero entries.
2. The first nonzero entry in each row is a 1.
3. In any two consecutive (nonzero) rows, the
leading 1 in the lower row is to the right of
the leading 1 in the upper row.
4. If a column contains a leading 1, then the
other entries in that column are zeros.
Row–Reduced Form of a Matrix
Row-Reduced Form
Non Row-Reduced Form
1 0 0 3 


0 1 0 1
0 0 1 2 
1 0 0 9 


0 0 1 4 
0 1 0 2 
1 0 0 8 


0 1 0 4 
0 0 0 0 
R2 , R3
switched
Must
be 0
1 0 5 1 


0 1 0 3 
0 0 1 5
Unit Column
A column in a coefficient matrix where one of the
entries is 1 and the other entries are 0.
1 0 5 1 


0
1
0
3


0 0 1 5
Unit columns
Not a Unit
column
Pivoting – Using a coefficient to transform a
column into a unit column
 1 1 1 2 



1
3

3
0


 2 1 1 3 
1 1 1 2 


0
2

2
2


0 3 1 1
This is called pivoting on the 1 and it is
circled to signify it is the pivot.
Gauss-Jordan Elimination Method
1. Write the augmented matrix
2. Interchange rows, if necessary, to obtain a
nonzero first entry. Pivot on this entry.
3. Interchange rows, if necessary, to obtain a
nonzero second entry in the second row. Pivot
on this entry.
4. Continue until in row-reduced form.