Section 2.2 Systems of Linear Equations: Unique Solutions

Augmented Matrices The system of equations

2x+ 4y � 8z = 22

3x� 8y + 5z = 27

x� 7z = 33

can be represented as the following matrix

2

642 4 �8

3 �8 5

1 0 �7

�������

22

27

33

3

75

Example 1: What value is in row 1, column 2 of the above matrix?

Example 2: Find the augmented matrix for the following system of equations.

9x+ 5y � 10z = 11

4x� 12y + 17z = 37

x� 2y = 45

Example 3: Find the system of equations for the following augmented matrix.

2

6410 0 �6

30 �9 0

1 19 �12

�������

29

31

10

3

75

Row-Reduced Form of a Matrix

1. Each row consisting entirely of zeros lies below all rows having nonzero entries

2. The first nonzero entry in each (nonzero) row is a 1 (called a leading 1).

3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading

1 in the upper row.

4. If a column in the coe�cient matrix has a leading 1, then the other entries in the column are

zeros.

Example 4: Which of the matrices below are in row-reduced form?

2

641 0 �6

0 1 8

0 0 0

�������

9

1

0

3

75

2

641 0 �6

0 0 0

0 1 �12

�������

2

0

6

3

75

Row Operations

1. Interchange any two rows.

2. Replace any row by a nonzero constant multiple of itself.

3. Replace any row by the sum of that row and a constant multiple of any other row.

Unit Column A column in a coe�cient matrix is called a unit column if one of the entries is a 1

and the other entries are zeros.

Note: If you transform a column in a coe�cient matrix into a unit column then this is called pivotting

on that column.

Notation for Row Operations Letting Ri denote the ith row of a matrix, we write:

Operation 1. Ri $ Rj Interchange row i with row j.

Operation 2. cRi to mean: Replace row i with c times row i.

Operation 3. Ri + aRj to mean: Replace row i with the sum of row i and a times row j.

2 Fall 2017, Maya Johnson

Example 5: Pivot the matrix below about the entry in row 1, column 1

"3 6

2 2

�����9

3

#

The Gauss-Jordan Elimination Method

1. Write the augmented matrix corresponding to the Linear system.

2. Interchange rows (Operation 1), if needed, to obtain an augmented matrix in which the first

entry in the first row is nonzero. Then pivot the matrix about this entry.

3. Interchange the second row with any row below it, if needed, to obtain an augmented matrix in

which the second entry in the second row is nonzero. Then pivot the matrix about this entry.

4. Continue until the final matrix is in row-reduced form.

Example 6: Solve the following system of linear equations using the Gauss-Jordan elimination method.

a)

2x+ 6y = 1

�6x+ 8y = 10

3 Fall 2017, Maya Johnson

b)

2x+ 2y = 4

�3x+ 6y = 5

c)

2x1 + x2 � x3 = 3

3x1 + 2x2 + x3 = 8

x1 + 2x2 + 2x3 = 4

Example 7: A person has four times as many pennies as dimes. If the total face value of these coins

is $1.26, how many of each type of coin does this person have?

4 Fall 2017, Maya Johnson

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Example 8: Cantwell Associates, a real estate developer, is planning to build a new apartment complex

consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 168 units is

planned, and the number of family units (two- and three-bedroom townhouses) will equal the number

of one-bedroom units. If the number of one-bedroom units will be 3 times the number of three-bedroom

units, find how many units of each type will be in the complex.

9 Fall 2017, Maya Johnson

X = # of one - bedroom units

y = # of two - bedroom units

z = # of three - bedroom units

" total 168 units"

⇒ xtytz = 168

"

# of family units equals # of one - bedroom units "

⇒ yt Z a ×

" # of one - bedroom units is 3 times three - bedroom units "

⇒ ×= zz

System of Linear Equations :

X ty tz = 168 X ty t Z = 168

y + Z = ×

⇒ → + y t z = O

× = ZZ×

- 3Z = 0

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28 three - bedroom units