Solving systems using matrices

4.6: Matrix equations

4.7: Augmented matrices

Solving systems using matrices

12

532

yx

yx

1

5

21

32

y

x

Matrix equation Augmented matrix

121

532

CoefficientsVariables

constants

Coefficients

constants

Matrix Equations

• Be sure that the system is lined up before you write matrices

• The first matrix is the coefficients of the variables

• The second matrix contains the variables (this matrix is only for notation)

• The third matrix is the constants (numbers after the equal sign)

Solve a matrix equation

• Enter the coefficient matrix in as matrix A• Enter the constant matrix in as B

• Type in

• The solution will show up as a 2x1 matrix

][][ 1 BA

EXAMPLE:

12

532

yx

yx

1

5

21

32

y

x

Examples:

396

2224 .2

113

185 .1

yx

yx

yx

yx

11

1

13

85

y

x

3

22

96

24

y

x

Augmented Matrices

• Be sure to line up the system first• Put all of the coefficients into one matrix (the #

of columns will be 1 more than the # of rows)• In the Matrix Math menu choose option B:rref(• Select matrix A and press enter• The result will be the identity matrix with the

solutions to the system in the last column

Example:

81332

1024

4322

zyx

zx

zyx

81332

10204

4322

No Solution and Many solutions

Matrix equations will result in an error message when the system is no solution OR many solutions

Augmented matrices can be identified as no solution or many solutions by looking at the bottom row of the solution

100

201

000

421

The identity matrix will be missing from the solution

The system has No Solution if the last row is all zeros with a one at the end of the row

The system has Many Solutions if the last row is all zeros

Examples:

3.35.03.0

8.12.02.0 .2

703510

28144 .1

yx

yx

yx

yx

703510

28144

3.35.03.0

8.12.02.0