5.2 systems of equations in three unknowns
18
Chapter 5.2 Matrices and Determinants 1
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Transcript of 5.2 systems of equations in three unknowns
Chapter 5.2 Matrices and Determinants
1
Matrix
11 12 1
21 22 2
1 2
An is a rectangular array of numbers
with rows a
matrix
nd columns
n
n
m m mn
m n
a a a
a a a
m n
a a a
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Determinant of a 2×2 Matrix
1 1
2 2
1 1
1 2 2 1
2 2
T determinanhe of the matrix
is given by
t
.
a bA
a b
a bA a b a b
a b
3
Example 5.2.1
10 8Find the determinant of .
1 5
10 850 8 42
1 5
M
M
4
Cramer’s Rule
1 1 1
2 2 2
1 1 1 1 1 1
2 2 2 2 2 2
Given the system with
, , and ,
the solution to the linear system is given by
and .
provided that 0.
x y
yx
a x b y c
a x b y c
a b c b a cA A A
a b c b a c
AAx y
A A
A
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Example 5.2.2
Solve the system using Cramer's Rule.
5 13
2 3 12
5 1 13 1 5 13
2 3 12 3 2 12
5 115 2 17
2 3
x y
x y
x y
A A A
A
6
5 1 13 1 5 13
2 3 12 3 2 12
17
13 139 12 51
12 3
5 1360 26 34
2 12
51 343 2
17 17
3,2
x y
x
y
x x
A A A
A
A
A
A Ax y
A A
SS
7
Nature of Solutions
If 0 then the system has a unique solution.
If 0 and at least one of and is
not zero, then the system is inconsistent.
If 0 and BOTH 0 and 0, then
the system is dependent.
x y
x y
A
A A A
A A A
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Determinant of a 3×3 Matrix
1 1 1
2 2 2
3 3 3
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
1 2 3 3 1 2 2 3 1 3 2 1 1 3 2 2 1 3
The of a 3 3 matrix
is given
determinant
by
a b c
A a b c
a b c
a b c a b
A a b c a b
a b c a b
a b c a b c a b c a b c a b c a b c
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Example 5.2.3
2 3 1
Find the determinant of 4 3 2 .
1 1 1
2 3 1 2 3
4 3 2 4 3
1 1 1 1 1
6 6 4 3 4 12
8 13 5
A
A
10
Cramer’s Rule
1 1 1 1
2 2 2 2
3 3 3 3
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
Given the system with
x
y z
a x b y c z d
a x b y c z d
a x b y c z d
a b c d b c
A a b c A d b c
a b c d b c
a d c a b d
A a d c A a b d
a d c a b d
11
Cramer’s Rule
Then the solution to the linear system is given by
provided that 0
yx zAA A
x y zA A A
A
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Nature of Solutions
If 0 then the system has a unique solution.
If 0 and at least one of , , and is
not zero, then the system is inconsistent.
If 0 and , , and are all zero, then
the system is dependent.
x y z
x y z
A
A A A A
A A A A
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Example 5.2.4
Solve the system using Cramer's Rule.
2 3
2
2 3
0 2 1 3 2 1
1 1 1 2 1 1
1 1 2 3 1 2
0 3 1 0 2 3
1 2 1 1 1 2
1 3 2 1 1 3
x
y z
y z
x y z
x y z
A A
A A
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Example 5.2.4
0 2 1 0 2
1 1 1 1 1
1 1 2 1 1
0 2 1 1 0 4 6
3 2 1 3 2
2 1 1 2 1
3 1 2 3 1
6 6 2 3 3 8 6
x
A
A
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6A 6xA
4yA 10zA
61
6xAxA
4 2
6 3
yAy
A
10 5
6 3zAzA
2 51, ,
3 3SS
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Exercises
Answer odd-numbered items in Leithold
Exercise 9.6, page 564 (1-43)
Check your answers on A-65.
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End of Chapter 5.2
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