10.2 Systems of Linear Equations: MatricesObjectives

1. Write the Augmented Matrix

2. Write the System from the Augmented matrix

3. Perform Row Operations

4. Solve a System of Linear Equations using Matrices

a) Row operations

b) Row-echelon form (Gauss Elimination)

c) Reduced Row-echelon (Gauss-Jordan)

1. Augmented MatrixGiven the system:

The coefficient matrix is:

The augmented matrix is:

10223

42

732

zyx

zyx

zyx

1. Augmented Matrixwrite the augmented matrix

4

93

1952

z

zy

zyx

2. Write System from augmented matrix

1) Write the system2) Now solve it !

2

5

1

200

160

532#1

2. Write System from augmented matrix

Write the system and solve.

5

8

4

100

110

021#2

3. Row OperationsNotation:

new row after row operations are applied

original row

Multiply row i by a constant k

Interchange row 1 and row 2

iriR

ikr

21 rr

Perform each operation (using the “previous” matrix for each)

1. (Interchange row 1 and row 2)

2. (Add row 3 to row 2)

3. (Add 3 times row 1 to row 3)

4.

3. a) An example of row operations

232 rrR 21 rr

10

4

7

223

111

323

313 3 rrR

32 rr

5. What

operation

would put 1 in

position 2,2 ?

3 b) Row-Echelon FormAugmented matrix reduced to a form with 1’s on diagonal

and 0’s beneath diagonal is called row-echelon form

A 2x2 system would have the form:

A 3x3 system would have the form:

The method for solving a system using row-echelon form is also known as Gaussian Elimination.

f

e

d

c

ba

100

10

1

c

ba

10

1

Note: The augmented matrix from warm-up is in row-echelon form

Example 1: Solve the 2x2 system p. 755 #37

Handout 10.2: Solving Linear Systems using Matrices

Review: Solve a system of 3 equations using Substitution

4

93

1952

z

zy

zyx