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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)
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1. Augmented MatrixGiven the system:
The coefficient matrix is:
The augmented matrix is:
10223
42
732
zyx
zyx
zyx
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1. Augmented Matrixwrite the augmented matrix
4
93
1952
z
zy
zyx
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2. Write System from augmented matrix
1) Write the system2) Now solve it !
2
5
1
200
160
532#1
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2. Write System from augmented matrix
Write the system and solve.
5
8
4
100
110
021#2
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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
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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 ?
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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
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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
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Review: Solve a system of 3 equations using Substitution
4
93
1952
z
zy
zyx