4.5 Solving Systems using 4.5 Solving Systems using Matrix Equations and Matrix Equations and

InversesInverses

4.5 Solving Systems using 4.5 Solving Systems using Matrix Equations and Matrix Equations and

InversesInverses

OBJ: To solve systems of linear equations using inverse matrices & use systems of linear equations to solve real-life problems

6

8

21

45

y

x

A linear system can be written as a matrix equation AX=B

Coefficient matrix Variable

matrix

Constant matrix

5 4 8

1 2 6

x y

x y

Ex 1: Write as a matrix equation

3 4 5

2 10

x y

x y

10

5

12

43

y

x

Suppose 8x = 24

How do you solve for x?

We cannot divide matrices, but we can multiply by the inverse.

AX = BAA-1-1 AA-1-1

IX = AA-1-1B

X = AA-1-1B

DO NOT COPY Solving Matrix Equations

Ex 2: Solve using matrices

3 4 5

2 10

x y

x y

x = -7

y = -4

10

5

12

43

y

x

A B

X = AA-1-1B

AX = B

(-7, -4)

Ex 3: Solve 7 3 11

14 4 2

x y

x y

x = 5/7

y = 2

(5/7, 2)

Ex 4: Solve using matrices

2 3 1

3 3 1

2 4 2

x y z

x y z

x y z

x = 2

y = -1

z = -2

(2, -1, -2)Ex 5: Solve using matrices

x y z

x y

x y z

3 5 15

2 1

9 8 4 12

x = 4

y = -7

z = 2

(4, -7, 2)