4.44.4Solving systems using matrices

““A” MatrixA” Matrix

A matrix is a rectangular array of numbers.

“The Matrix” is a movie with Keanu Reeves.

““The Matrix”The Matrix”

Example of a matrixExample of a matrix

29

64

Columns

Rows

Element

Note: A Square matrix has the same # of rows and columns

Writing an Augmented MatrixWriting an Augmented Matrix

Linear Equations 1:

Linear Equations 2:

Augmented Matrix

111 cybxa 222 cybxa

222

111

cba

cba

Note these are Standard Form

Writing an Augmented MatrixWriting an Augmented Matrix

Linear Equations 1:

Linear Equations 2:

Augmented Matrix

1035 yx1297 yx

???

???

Note these are Standard Form

EX. 1

Writing an Augmented MatrixWriting an Augmented Matrix

Linear Equations 1:

Linear Equations 2:

Augmented Matrix

1795 xyxy 8142

???

???

Write in Standard Form!!!

EX. 2

Row Transformations

All numbers in a row may be multiplied or divided by any nonzero real number.

You can replace rows by adding them to other rows and placing the sum in the row.

Transformations Example 1All numbers in a row may be multiplied or divided by any nonzero real number.

380

243

Multiply R1 by -2

=

380

486

Transformations Example 2All numbers in a row may be multiplied or divided by any nonzero real number.

339

117

Divide R2 by 3

=

???

???

Example 2 ANSWER

113

117

Transformations Example 3All numbers in a row may be multiplied or divided by any nonzero real number.

1032

304

Multiply R1 by 2 and multiply R2 by -4

=

???

???

Example 3 ANSWER

40128

608

Transformations Example 4You can replace rows by adding them to other rows and placing the sum in the row.

635

412

Replace R1 with R1+R2

=

635

1023

Transformations Example 5You can replace rows by adding them to other rows and placing the sum in the row.

134

7126

Replace R2 with R1-R2

=

???

???

Example 5 ANSWER

8152

7126

Transformations Example 6

111014

1247

Replace R1 with : -2R1 + R2

=

???

???

Example 6 ANSWER

111014

35180

Note: R2 does not change!!!!

Transformations Example 7

4810

145

Replace R2 with : -1/2R2 – R1

=

???

???

Example 7 ANSWER

Note: R1 does not change!!!!

300

145

Triangular formTriangular form

q

pa

10

1

The 1’s and the 0 in these locations

a, p, and q are just constants

Use row transformation to get a matrix in triangular form

1.Work in column 1 to get the one.

2. Get the zero in column 1.

3. Get the 1 in column 2.

q

pa

10

11st

2nd

3rd

Triangular form Example 1

4810

24126

Write the matrix in Triangular form

=

???

???

Example 1 Steps

1st : 1/6 R12nd : Replace R2 with 10R1 + R23rd : -1/28 R2

Let’s Look at it !

Example 1 ANSWER

310

421

Triangular form Example 21. Write the Linear Equations in standard form. 2. Write the Augmented Matrix.3. Get the matrix in Triangular Form.4. Write the matrix back into Standard form.5. Solve for x and y.

yx

xy

252

2103

1.1. Put in Standard form.Put in Standard form.

2. Write the Augmented Matrix2. Write the Augmented Matrix

???

???

yx

xy

252

2103

522

1032

yx

yx

522

1032

3. Try for Triangular Form.3. Try for Triangular Form.

4. Back to Standard Form.4. Back to Standard Form.

522

1032

110

5231

110

5231

110

10231

yx

yx

5. Solve for x and y.5. Solve for x and y.

110

10231

yx

yx

Looking here.

y = -1, now substitute into equation 1.

x = 7/2 Therefore ( 7/2 , -1)is where the lines cross.