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example 1 Matrix Solution of Linear Systems
Chapter 7.2
Use matrix row operations to solve the system of equations
2 3 1
2 3
3 9
x y z
x y z
x y z
2009 PBLPathways
2009 PBLPathways
Use matrix row operations to solve the system of equations
2 3 1
2 3
3 9
x y z
x y z
x y z
2009 PBLPathways
Use matrix row operations to solve the system of equations
2 3 1
2 3
3 9
x y z
x y z
x y z
2 3
2 3 1
3 9
x y z
x y z
x y z
R1 R2
2 3 1 1
1 1 2 3
3 1 1 9
1 1 2 3
2 3 1 1
3 1 1 9
2009 PBLPathways
Use matrix row operations to solve the system of equations
2 3 1
2 3
3 9
x y z
x y z
x y z
2 3
2 3 1
3 9
x y z
x y z
x y z
R1 R2
2 3 1 1
1 1 2 3
3 1 1 9
1 1 2 3
2 3 1 1
3 1 1 9
2009 PBLPathways
Use matrix row operations to solve the system of equations
2 3 1
2 3
3 9
x y z
x y z
x y z
2 3
2 3 1
3 9
x y z
x y z
x y z
R1 R2
2 3 1 1
1 1 2 3
3 1 1 9
1 1 2 3
2 3 1 1
3 1 1 9
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
2 2 4 6
2 3 1 1
0 1 3 5
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
2 2 4 6
2 3 1 1
0 1 3 5
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
2 2 4 6
2 3 1 1
0 1 3 5
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
4 7 18
x y z
y z
y z
-1 R2 R2 1 1 2 3
0 1 3 5
0 4 7 18
-2 R1 + R2 R2-3 R1 + R3 R3
2 3
3 5
4 7 18
x y z
y z
y z
1 1 2 3
0 1 3 5
0 4 7 18
1 1 2 3
2 3 1 1
3 1 1 9
3 3 6 9
3 1 1 9
0 4 7 18
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38
2009 PBLPathways
2 3
3 5
19 38
x y z
y z
z
-4 R2 + R3 R3
2 3
3 5
2
x y z
y z
z
R3 R3119
1 1 2 3
0 1 3 5
0 0 19 38
1 1 2 3
0 1 3 5
0 0 1 2
1 1 2 3
0 1 3 5
0 4 7 18
0 4 12 20
0 4 7 18
0 0 19 38