4.5: Solving Systems Using Inverse MatricesObjective:Students will solve systems of equations using inverse matrices.

Investigating Matrix Equations, p. 230.

A linear system can be written as a matrix equation:

AX = BA= Coefficient Matrix

X = Matrix of VariablesB= Matrix of Constants

Write the following as a matrix equation:

123

1952

yx

yx

You can solve for the Matrix of Variables by multiplying EACH side of the matrix by A-1 on the left.

BAX

BAIX

BAAXA

1

1

11

Solve:

1

19

23

52

y

x

Examples: Write and solve the following using matrices.

1. 2.

113

132

yx

yx

122

824

yx

yx

Systems with 3 variables…

• Solution: (x, y, z) Graph: (-2, 3, -4)• 3 different

unknowns in a system of 3 variables

Use a matrix equation to solve.

2043

34254

1432

zyx

zyx

zyx

Augmented Matrices: contains coefficient matrix and matrix of constants

3152

2642

3152

2642

yx

yx• Matrix-MATH-rref-

enter –quit-matrix, call up the matrix you want

• rref([A])(rref= reduced row echelon form)

• Answers are in the last column. Other rows contain 1’s along main diagonal and 0’s below

In the 1968 presidential election, 538 electoral votes were cast. Of these x went to Richard Nixon, y went to Hubert Humphrey and z went to George Wallace. The value of x is 110 more than y. The value of y is 145 more than z. Write a system of equations and solve to find out how many votes each candidate received.

The senior class is selling boxed greeting cards. Birthday cards sell for $5.00 a box, while thank you cards sell for $7.00 a box. You sold 4 more boxes of birthday cards than thank you cards; your total sales amounted to $152. How many boxes of each kind did you sell? Write a system of equations and solve using inverse matrices.

You have $10,000 to invest in two types of stock. The expected annual returns for the stocks are; 10% for stock A and 6% for stock B. You want the overall annual return to be 8%. Write a linear system of equations that represents the given information. Write a matrix equation and solve to find out how much you should invest in each type of stock.