Using Matrices to Solve Systems of Equations

Matrix Equations

- We have solved systems using graphing, but now we

learn how to do it using matrices. This will be

particularly useful when we have equations with

three variables.

Matrix Equation

- Before you start, make sure that both of your

equations are in standard form and the variables

are in the same order (alphabetical usually is

best).

Setting up the Matrix Equation

- Given a system of equations -2x - 6y 0

3x 11y 4 - Since there are 2 equations, there will be 2

rows. - Since there are 2 variables, there will be 2

columns.

Setting up the Matrix Equation

- There are 3 parts to a matrix equation
- 1)The coefficient matrix,
- 2)the variable matrix, and
- 3)the constant matrix.

-2x - 6y 0 3x 11y 4

- The coefficients are placed into the coefficient

matrix.

-2x - 6y 0 3x 11y 4

- Your variable matrix will consist of a column.

-2x - 6y 0 3x 11y 4

- The matrices are multiplied and represent the

left side of our matrix equation.

-2x - 6y 0 3x 11y 4

- The right side consists of our constants. Two

equations two rows.

-2x - 6y 0 3x 11y 4

- Now put them together.

Well solve it later!

Create a matrix equation

- 3x - 2y 7 y 4x 8
- Put them in Standard Form.
- Write your equation.

Create a matrix equation

- 3a - 5b 2c 9
- 4a 7b c 3
- 2a - c 12

Solving a matrix equation

- To solve matrix equations, get the variable

matrix alone on one side. - Get rid of the coefficient matrix by multiplying

by its inverse

- When solving matrix equations we will always

multiply by the inverse matrix on the left of the

coefficient and constant matrix. (remember

commutative property does not hold!!)

- The left side of the equation simplifies to the

identity times the variable matrix. Giving us

just the variable matrix.

- Using the calculator we can simplify the left

side. The coefficient matrix will be A and the

constant matrix will be B. We then find A-1B.

- The right side simplifies to give us our answer.
- x -6
- y 2
- You can check the systems by graphing,

substitution or elimination.

Advantages

- Basically, all you have to do is put in the

coefficient matrix as A and the constant matrix

as B. Then find A-1B. This will always work!!! - NO SOLVING FOR Y!!!!! )

Solve

- Plug in the coeff. matrix as A
- Put in the const. matrix as B
- Calculate A-1B.

Solve

- r - s 3t -8
- 2s - t 15
- 3r 2t -7

Word Problem Systems

- The sum of three numbers is 12. The 1st is 5

times the 2nd. The sum of the 1st and 3rd is 9.

Find the numbers.

Word Problem Systems

- The sum of three numbers is 12.
- x y z 12
- The 1st is 5 times the 2nd.
- x 5y
- The sum of the 1st and 3rd is 9.
- x z 9

Word Problem Systems

- x y z 12 x 5y gt x - 5y 0 x z

9

Word Problem Systems

- x y z 12 x - 5y 0 x z 9