Download - 4.7 Inverse Matrices and Systems. 1) Inverse Matrices and Systems of Equations You have solved systems of equations using graphing, substitution, elimination…oh.

4.7 Inverse Matrices and Systems

1) Inverse Matrices and Systems of Equations

You have solved systems of equations using graphing, substitution, elimination…oh my…

In the “real world”, these methods take too long and are almost never used.

Inverse matrices are more practical.


For a System of Equations

1453

52

yx

yx


For a We can write a System of Equations Matrix Equation

1453

52

yx

yx

14

5

53

21

y

x


Example 1:Write the system as a matrix equation

62

1132

yx

yx



Matrix Equation

62

1132

yx

yx

6

11

21

32

y

x



Matrix Equation

62

1132

yx

yx

6

11

21

32

y

x

Coefficient matrix

Constant matrix

Variable matrix


Example 2:

822

52

0

zyx

zyx

zyx


Example 2:

8

5

0

212

121

111

z

y

x

822

52

0

zyx

zyx

zyx


Example 2:

A BX

8

5

0

212

121

111

z

y

x

822

52

0

zyx

zyx

zyx


BAX


BAX 1

When rearranging, take the inverse of A

BAX


The Plan…

“Solve the system” using matrices and inverses

BAX 1BAX


Example 3:Solve the system

62

1132

yx

yx



Step 1: Write a matrix equation

62

1132

yx

yx



Step 1: Write a matrix equation

6

11

21

32

y

x

62

1132

yx

yx



Step 2: Find the determinant and A-1

62

1132

yx

yx




21

32A

62

1132

yx

yx

Change signs

Change places




21

32A

62

1132

yx

yx

Change signs

Change places

detA = 4 – 3

= 1




21

32

21

32

1

1

21

32

1

1

A

A

A

62

1132

yx

yx



Step 3: Solve for the variable matrix

62

1132

yx

yx




BAy

x

BAX

1

1

62

1132

yx

yx




1

4

6

11

21

32

1

1

y

x

y

x

BAy

x

BAX

62

1132

yx

yx




1

4

6

11

21

32

1

1

y

x

y

x

BAy

x

BAX

The solution to the system is (4, 1).

62

1132

yx

yx


Example 4:Solve the system. Check your answer.

523

735

ba

ba



5

7

23

35

b

a

523

735

ba

ba



523

735

ba

ba

53

32

53

32

1

1

23

35

1

1

A

A

A

detA = 10 - 9

= 1



523

735

ba

ba

4

1

5

7

53

32

1

1

b

a

b

a

BAb

a

BAX



523

735

ba

ba

The solution to the system is (-1, 4).

4

1

5

7

53

32

1

1

b

a

b

a

BAb

a

BAX



Check

523

735

ba

ba

77

7125

7)4(3)1(5

735

ba

55

583

5)4(2)1(3

523

ba

What about a matrix that has no inverse?

It will have no unique solution.



Example 5:Determine whether the system has a unique solution.

842

52

yx

yx



Find the determinant.

842

52

yx

yx




842

52

yx

yx

8

5

42

21

y

x




0

)2(2)4(1

42

21det

42

21

A

A

842

52

yx

yx

Since detA = 0, there is no inverse.

The system does not have a unique solution.

Homework

p.217 #1-5, 7-10, 20, 21, 23, 24, 26, 27, 36

DUE TOMORROW: Two codes

TEST: Wednesday Nov 25Chapter 4