4.74.7Solving Systems using Solving Systems using Matrix Equations and Matrix Equations and

InversesInverses

Matrix Equation

68

2145yx

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

Coefficient matrix Variable

matrix

Constant matrix

5 4 81 2 6x yx y

Ex. 1 Write as a matrix equation.

3 4 52 10x yx y

105

1243

yx

Suppose ax = bHow 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-1-1BX = AA-1-1B

Solving Matrix Equations

Ex. 2 Solve using matrices.

3 4 52 10x yx y

x = -7y = -4

105

1243

yx

A B

X = AA-1-1BAX = B

(-7, -4)

Ex. 3 Solve using matrices7 3 1114 4 2x yx y

x = 5/7y = 2

(5/7, 2)

Ex. 4 Solve using matrices2 3 13 3 12 4 2

x y zx y zx y z

x = 2y = -1z = -2

(2, -1, -2)

Ex. 5 Solve using matrices

x y zx yx y z

3 5 152 19 8 4 12

x = 4y = -7z = 2

(4, -7, 2)

Unique Solutions• Find detA. If it = 0 then there is

an unique solution.

• If detA = 0 then the system does not have a unique solution.

Determine whether each system has an unique solution.

20x + 5y = 14530x – 5y = 125

Assignment

Pg. 213 1-21 odd